MODELING OF INTERACTIONS BETWEEN NANOPARTICLES AND CELL MEMBRANES

Transcription

1 MODELING OF INTERACTIONS BETWEEN NANOPARTICLES AND CELL MEMBRANES By YOUNG-MIN BAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

2 c 2010 Young-Min Ban 2

3 To my encouraging parents; my princess, Suh-Eu; and my loving husband, Hyung-Seok 3

4 ACKNOWLEDGMENTS It is a pleasure to thank to my advisor, Dr. Dmitry I. Kopelevich, for instilling in me the qualities of being a self-motivated researcher. I owe my deepest gratitude to him for giving me the opportunity to develop my own individuality and self-sufficiency by being allowed to work with such independence. His guidance and support from the preliminary to the concluding level enabled me to develop an understanding of the field of molecular modeling. His constant patience and encouragement helped me overcome many crisis situations and finish this dissertation. I hope that one day I would become as good an advisor to my students as Dr. Kopelevich has been to me. I am also deeply grateful to the members of my committee, Dr. Aravind Asthagiri, Dr. Yiider Tseng, and Dr. Bonzongo for their valuable suggestions and advice regarding my research. A very special thank you to my friend Beverly Hinojosa for giving me valuable advice. I am thankful to her for reading my reports and commenting on my views. I also thank my research group members, including Chia-Yi Chen, Gunjan Mohan, Ashish Gupta, and Young-Nam Ahn for their support. Most importantly, none of this would have been possible without the love and patience of my husband Hyung-Seok. His support, encouragement, quiet patience and unwavering love have shaped me to be the person I am today. I would like to express my heart-felt gratitude to him. Finally, I appreciate the financial support from Environmental Protection Agency and National Science Foundation that funded the research discussed in this dissertation. I acknowledge University of Florida High-Performance Computing Center for providing computational resources. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Background Specific Aims Overview of Dissertation METHODS Molecular Dynamics Simulations Coarse-Grained Molecular Model Stochastic Model and CMF Method Calculation of Elastic Properties of Membrane Estimation of Bending and Tilt Modulus Statistical Analysis of Correlated Time Series Theory Numerical algorithm Error estimation Validation Stress Tensor TRANSPORT OF CARBON-BASED NANOPARTICLES THROUGH LIPID MEMBRANES Introduction Free Energy Profiles Effect of Fullerenol Orientation on Membrane Deformation Local Membrane Energy and Effective Interaction Potential between Nanoparticles Conclusions ASSESSMENT OF POSSIBLE NEGATIVE EFFECTS OF CNTS ON LIPID MEMBRANES Introduction Model and Simulation Details System Structure

6 4.4 Elastic Properties of Membrane Effect of CNT on Pressure Distribution inside Membranes Membrane Energy around Nanoparticles Conclusions LIPID PEROXIDATION Introduction Model and Simulation Details Effect of Lipid Peroxidation on Membrane Properties Effect of Nanoparticles on Peroxidized Lipid Bilayers Conclusions CONCLUSIONS REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 4-1 Box sizes of equilibrated systems The average fraction of folded lipids and the average area per lipid in bilayers with various concentrations of peroxidized lipids The average area per lipid in DPPC-25% and DPPC-75% with and without a fullerenol

8 Figure LIST OF FIGURES page 2-1 CG models of DPPC lipid molecule, fullerene (left) and fullerenol (right), and CNT Self-assembly of DPPC bilayer Cross-sections of slabs and columns in which the stress tensor is averaged Density profile of the DPPC bilayer and free energy profiles for fullerene (solid line) and fullerenol (dashed-dotted line) Dependence of the diffusivity of fullerene on the nanoparticle position within the bilayer Local deformation of a lipid membrane containing a fullerene or a fullerenol Average dividing surfaces corresponding to the fullerenol constrained at Þ = -4.0 nm (dash dotted line, gray circle), Þ = -2.7 nm (dashed line, hollow circle) and Þ = 0.0 nm (solid line, black circle) Dependence of the Fourier mode with wavenumber Õ = 0.46 nm ½ of the dividing surface of the leaflet containing the nanoparticle on the nanoparticle position Þ Envelops of ACF of the random force acting on the fullerenol constrained at z = -4.0 nm (dashed-dotted line), z = -2.7 nm (dashed line), and z = 0.0 nm (solid line) Correlation times of the slowest fluctuations of the random force acting on particles Definition of the fullerenol orientation: angle between C ¼ (OH) ½¼ director (d) and the z-axis and contribution of the fullerenol orientation to the free energy (kj/mol) A schematic of free energy profile for fullerenol and its orientational behavior Average height of dividing surfaces of the bilayer containing a fullerenol located at the center oriented at 0 degree and 90 degrees with respect to z-axis Comparison of local average heights of lipids around nanoparticles Effect of fullerenol embedded in a DPPC membrane on upper and lower membrane leaflet Energy (kj/mol) of membrane containing a fullerenol

9 4-1 Lateral pressure within membrane and corresponding different conformational states of a hypothetical membrane protein Molecular model of a DPPC lipid bilayer containing a carbon nanotube Probability distributions of distance Þ between the bilayer and CNT centers of mass and orientation Ó θ of CNT with respect to the Þ-axis Spectral intensity of fluctuation of membrane undulations and lipid tilt in pure DPPC membrane, as well as DPPC membranes containing CNT of various length (see legend) at concentration ¼º¼½ CNT/nm ¾ Dependence of the HK model parameters κ and κ Ø on the upper cut-off wavelength Õ Ñ Ü Distributions of lateral pressure È in direction normal to the bilayer surface Distributions of lateral pressure È (in bar) in the bilayer plane for bilayers containing CNT, CNT, and CNT nanotubes Effect of CNT embedded in a DPPC membrane on upper and lower membrane leaflet Effect of CNT embedded in a DPPC membrane on upper and lower membrane leaflet Effect of CNT embedded in a DPPC membrane on upper and lower membrane leaflet Simulation snapshots of DPPC bilayers at 800ns 0% (pure DPPC lipid bilayer), DPPC-25%, and DPPC-50%, and DPPC-75% Spectral intensity of bilayer surface fluctuations of the pure DPPC bilayer and of DPPC-25%, DPPC-50%, and DPPC-75% Density profiles of head and tail groups of lipid and water in the pure DPPC bilayer, DPPC-25%, DPPC-50%, and DPPC-75% Density profiles of the phosphate beads (PO, dotted) of the head group, the glycerol (dash-dotted), the terminal beads (-CH, dashed), and the peroxidized terminal beads (-COOH, solid) in DPPC-25% Density profiles of the peroxidized terminal beads in DPPC-25% (dotted), DPPC-50% (dash-dotted), and DPPC-75% (solid) Peroxidized DPPC lipid with folded tails Density profiles of head (thin lines) and tail (thick lines) groups in DPPC-25% and DPPC-75% with and without fullerenol

10 5-8 Density profiles of head (thin lines) and tail (thick lines) groups in DPPC-25% and DPPC-75% with and without fullerenol Time evolution of the Fourier modes of DPPC-25% and DPPC-75% Spectral intensity of bilayer surface fluctuations for DPPC-25% and DPPC-75% with and without fullerenol

11 Abstract of dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MODELING OF INTERACTIONS BETWEEN NANOPARTICLES AND CELL MEMBRANES Chair: Dmitry I. Kopelevich Major: Chemical Engineering By Young-Min Ban August 2010 Rapid development of nanotechnology and ability to manufacture materials and devices with nanometer feature size leads to exciting innovations in many areas including the medical and electronic fields. However, the possible health and environmental impacts of manufactured nanomaterials are not fully known. Recent experimental reports suggest that some of the manufactured nanomaterials, such as fullerenes and carbon nanotubes, are highly toxic even in small concentrations. The goal of the current work is to understand the mechanisms responsible for the toxicity of nanomaterials. In the current study coarse-grained molecular dynamics simulations are employed to investigate the interactions between NPs and cellular membranes at a molecular level. One of the possible toxicity mechanisms of the nanomaterials is membrane disruption. Possibility of membrane disruption exposed to the manufactured nanomaterials are examined by considering chemical reactions and non-reactive physical interactions as chemical as well as physical mechanisms. Mechanisms of transport of carbon-based nanoparticles (fullerene and its derivative) across a phospholipid bilayer are investigated. The free energy profile is obtained using constrained simulations. It is shown that the considered nanoparticles are hydrophobic and therefore they tend to reside in the interior of the lipid bilayer. In addition, the dynamics of the membrane fluctuations is significantly affected by the nanoparticles at the bilayer-water interface. The hydrophobic 11

12 interaction between the particles and membrane core induces the strong coupling between the nanoparticle motion and membrane deformation. It is observed that the considered nanoparticles affect several physical properties of the membrane. The nanoparticles embedded into the membrane interior lead to the membrane softening, which becomes more significant with increase in CNT length and concentration. The lateral pressure profile and membrane energy in the membrane containing the nanoparticles exhibit localized perturbation around the nanoparticle. The nanoparticles are not likely to affect membrane protein function by the weak perturbation of the internal stress in the membrane. Due to the short-ranged interactions between the nanoparticles, the nanoparticles would not form aggregates inside membranes. The effect of lipid peroxidation on cell membrane deformation is assessed. The peroxidized lipids introduce a perturbation to the internal structure of the membrane leading to higher amplitude of the membrane fluctuations. Higher concentration of the peroxidized lipids induces more significant perturbation. Cumulative effects of lipid peroxidation caused by nanoparticles are examined for the first time. The considered amphiphilic particle appears to reduce the perturbation of the membrane structure at its equilibrium position inside the peroxidized membrane. This suggests a possibility of antioxidant effect of the nanoparticle. 12

13 CHAPTER 1 INTRODUCTION 1.1 Background Nanoparticles are defined as materials whose length is less than 100 nanometers. Chemical and physical properties of nanoparticles are often different from those of their bulk counterparts, which promotes their application in various fields. One possible application is the use of nanoparticles as a drug carrier, since nanoparticles are similar in size to biological molecules (proteins, DNA, and RNA) and are readily transported through the body due to their small size. Successful prototypes of drug delivery vehicles were developed using various types of nanoparticles, including carbon nanotubes (CNTs), fullerenes, and polymeric nanoparticles [17, 55]. Because of the growing number of applications, the production of nanomaterials is expected to significantly increase. Large amounts of manufactured nanoparticles might be released into the environment possibly affecting living organisms in a variety of ways. In particular, humans may be easily affected by nanoparticles because human skin and lungs are always in direct contact with the environment. Adverse effects of various nanoparticles on living organisms have been demonstrated by numerous recent studies. For example, pulmonary exposure of mice [21] and rats [58] to single-wall carbon nanotubes (SWCNT) has been shown to lead to lung inflammation. In addition to pulmonary toxicity, SWCNTs may cause dermal toxicity as indicated by studies of human skin cell culture exposed to SWCNTs [48]. Apparent toxicity of other carbon-based nanoparticles such as fullerenes has also been observed. Exposure to an aqueous solution of low concentration (0.5 ppm) of uncoated fullerenes leads to significant oxidative brain damage of fish within 48 hours [36]. Moreoever, the presence of fullerenes at concentrations as low as 20 ppb induces membrane rupture and eventual death of human dermal fibroblast cells [43]. 13

14 The main goal of this work is to investigate possible mechanisms of cell membrane damages due to its interactions with carbon-based nanoparticles (such as fullerenes and CNTs). One of the proposed mechanisms of the cell disruption involves the particle-induced oxidative stress and the corresponding lipid peroxidation [36, 43, 44, 47]. Reactive oxygen species (ROS), which are precursors to the peroxidation reaction were observed in cells exposed to several types of nanoparticles including SWCNT and fullerenes [43]. The ROS produces unstable lipid radicals, which react with other lipids eventually resulting in the membrane peroxidation, followed by membrane disruption. In addition to this chemical mechanism, non-reactive physical interactions between nanoparticles and biological membranes have been considered as an alternative mechanism of membrane instability. Although details of this physical mechanism is currently not clear, some computational work has explored the interactions between nanoparticles and cell membranes at a molecular level. Molecular dynamics studies have shown translocation of nanoparticles through membranes [5] and the effects of the embedded nanoparticles on the properties of the membrane [59]. It is observed that carbon-based nanoparticles, such as fullerenes and some of fullerene derivatives have can easily permeate into the membrane interior and will stay inside the membrane for a long time. During the stay, nanoparticles may significantly perturb the membrane and even disrupt the membrane integrity. This study represents a step toward development of a fundamental understanding of the molecular mechanisms of interaction of manufactured nanoparticles with cellular membranes. Molecular dynamics (MD) simulations are performed to investigate trasport of nanoparticles into the cellular membrane interior, changes of the membrane properties due to nanoparticles absorbed inside the membrane interior, and breakdown of membrane integrity due to the lipid peroxidation. 14

15 1.2 Specific Aims The goal of this study is to understand interactions between nanoparticles and cell membranes at a molecular level. The interaction between carbon-based nanomaterials and membranes are focused on, modeled as phospholipid bilayers. Possible toxic effects of nanoparticles with various physicochemical properties are investigated, namely their size, shape and functional groups on the surface. The specific aims of this research are: Investigate the transport mechanism of nanoparticle across a membrane and evaluate of nanoparticle partition in the membrane interior. The free energy barrier for the transport of carbon-based nanoparticles across bilayers are computed using the constrained mean force method and examine the transport rate of carbon-based nanoparticles for the cell membrane permeation. Examine effects of nanoparticles on physical properties of membranes and investigate possible impact of nanoparticles on membrane stability and functionality of membrane proteins. The membrane physical properties are characterized by bending and tilt modulus and pressure. Evaluate influence of ROS and resultant membrane peroxidation induced by nanoparticles. To examine the dose-dependency of nanoparticles on toxicity, the concentration of peroxidized lipids are altered. 1.3 Overview of Dissertation In chapter 2, methods used to simulate bilayer-nanoparticle systems and analyze the simulations are presented. Coase-graned molecular model is used to mimic bilayer-nanoparticle systems. It is assumed that anoparticle transport can be described by the Langevin equation. Simulation of nanoparticle transport is performed using the Constrained Mean Force (CMF) method. Calculations of elastic properties of membrane and stress tensor inside membrane are given in this chapter. In chapter 3, transport mechanism of nanoparticle through a lipid membrane are discussed and effect of the nanoparticle on membrane deformation during the transport are examined. The considered nanoparticles are hydrophobic and therefore they tend to 15

16 reside in the interior of the lipid bilayer. It is observed that nanoparticle transport through the bilayer affects the dynamics of membrane fluctuations. In chapter 4, non-reactive physical mechanisms are considered to assess possible influence of embedded nanoparticle into a membrane on disruption of membrane integrity. It is observed that these nanoparticles may affect several physical properties of the membrane, including its bending modulus and the lateral pressure profile. In chapter 5, Chemical mechanisms are considered to examine effects of lipid peroxidation on membrane disruption. Changes in internal structure of membrane by the chemical mechanisms are demonstrated. In addition, effects of nanoparticle on the structure of membrane are assessed. In chapter 6, the dissertation and present its broader impacts are concluded. 16

17 CHAPTER 2 METHODS 2.1 Molecular Dynamics Simulations The system containing a lipid bilayer and carbon-based nanoparticles is modeled by molecular dynamics (MD) simulation. MD simulation is a technique for computing the properties of a microscopic system based on the classical mechanics. In principle, the purpose of using MD simulation is to obtain a trajectory of all molecules from an initial set of atom positions and atom velocities. The motion of all atoms is described by the Newton s equation of motion: Ñ ¾ Ö Ø ¾ ½ ººº ƺ (2 1) The forces are the negative partial derivatives of a potential function Í Ö ½ Ö ¾ ººº Ö Æ µ, Í Ö º (2 2) Here, refers to the particle number, Ö is the position, Ñ is the mass of the particle, and is the force acting on the particle. The potential function used in this work is described in section 2.2. The evolution of the atom positions and velocities is obtained by numerical integration of Eq. 2 1 using the Verlet leap-frog algorithm [1]. This algorithm uses the particle positions Ö at time Ø and velocities Ú at time Ø Ø ¾ Ø Ø and velocity at time Ø Ø ¾ ( Ú Ø Ø ¾ to update the position at using the force ص acting on the particle at time Ø, ) ( Ú Ø Ø Ö Ø Øµ Ö Øµ Ú ¾ ( ) ص Ñ Ø Ø ¾ Ø (2 3) ) غ (2 4) Newton s equations of motion imply that the total energy of the system remains constant but the kinetic energy of the particles changes, leading to temperature fluctuations. Since temperature is usually constrained in experimental systems, it is 17

18 desirable to maintain constant temperature in MD simulations. Several thermostating techniques are available to accomplish this goal and in this work the extended-ensemble approach first proposed by Nosé [34] and later modified by Hoover [14] (Nosé-Hoover thermostat) is used. The idea of this method is to extend the system by introducing a thermal reservoir. A friction term Ö is introduced in the equations of motion, ¾ Ö Ø ¾ Ñ ξ Ö Ø º (2 5) The dynamics of the friction coefficient ξ is governed by the follwing equation: ξ Ø ½ É Ì Ì ¼µº (2 6) Here, Ì and Ì ¼ are the current and reference temperatures of the system, respectively and the reservoir mass parameter É determines the strength of coupling with the thermal reservoir. The pressure in the system with constant volume also fluctuates. In this simualtion the pressure is kept constant by Parrinello-Rahman pressure coupling [35, 39], which is similar to the Nose-Hoover temperature coupling, ¾ Ö Ø ¾ Ñ Å Ö Ø (2 7) Å ½ [ Ø Ø ] ½ º (2 8) Here, Å is the friction coefficient, Î is the volume of the simulation box, and the matrix obeys the following equation of motion ¾ Ø Î ¾ Ï ½ ½ È È Ö µº (2 9) The coupling strength is determined by the mass parameter matrix Ï. The matrices È and È Ö are the current and reference pressures, respectively. Implementation of the T- and P-coupling schemes requires calculation of instantaneous T and P. Temperature of an N-particle system is given by its total kinetic 18

19 energy, ¾ Ì Ò ¾ Æ ½ Ñ Ú ¾ (2 10) where is the Boltzmann constant, Ñ is mass of i-th particle, and Ú is velocity of i-th particle. The pressure is calculated from the difference between kinetic energy Ò and the virial, È ¾ Î Ò µ (2 11) where the virial tensor is defined as ½ ¾ Ö º (2 12) < Here, Ö is the distance between the -th and -th particles and is the force applied by the -th particle on the -th particle. In this MD simulations periodic boundary conditions are used to minimize the edge effects which would be introduced by the presence of walls. When periodic conditions are used, molecules exiting from the simulation box on one side reappear by entering on the opposite side of the box. The molecules interact only with the nearest periodic images of other molecules. All MD simulations are performed using the GROMACS software package [54]. 2.2 Coarse-Grained Molecular Model In this study the Coarse Grained (CG) model proposed by Marrink et al. [28] is used. This model approximates small groups of atoms, usually four heavy atoms, as a single united atom. For example, four methyl/methylene groups are modeled by a single hydrophobic bead and four water molecules are modeled by a single hydrophilic bead. Because of their small size and mass, hydrogen atoms are not considered at all. This significantly reduces the computational time due to reduction in the number of degrees of freedom and smoother potentials which allow one to use larger timesteps. Despite the introduced simplification, this CG model has shown good agreement with experiments 19

20 and atomistic simulations for various lipid systems [3, 28, 32]. However, the CG model has some limitations. While some variables, such as densities, length scales, energies, temperature and pressure, can be accurately reproduced by the CG model, this is not exactly true for the time scale. The effective time in the CG model is 3-6 times larger because the CG interactions are much smoother compared to atomistic interactions. In addition, fine chemical details are inaccessible in any CG approach. Four main types of interaction sites are considered in this model: polar (È), nonpolar (Æ), apolar ( ) and charged (É). Polar sites represent neutral groups of atoms that would easily dissolve in water, apolar sites represent hydrophobic moieties, and nonpolar groups are used for mixed groups which are partly polar, partly apolar. Charged sites (É) are reserved for ionized groups. A number of subtypes for each particle type is considered to allow a more accurate representation of the chemical nature of the underlying atomic structure. Subtypes within each main type are distinguished by their hydrogen-bonding capabilities (d = donor, a = acceptor, da = both, 0 = none) and the degree of polarity. The nonbonded interactions between CG beads and are described by the Lennard-Jones (LJ) potential, [ ] (σ ) ½¾ ( σ ) Í ÄÂ Ö µ ǫ º (2 13) Ö Ö Here, σ is the effective minimum distance of approach between two particles and ǫ is the strength of their interaction. Ten levels of interactions are defined and the interaction strength ǫ of each of the interaction levels is as follows: the most polar interaction (O, ǫ = 5.6kJ/mol), attractive interactions (I, ǫ = 5.0kJ/mol), semi-attractive interactions in more volatile liquids such as ethanol or acetone (II, ǫ = 4.5kJ/mol, and III, ǫ = 4.0kJ/mol), intermediate interaction in aliphatic chains (IV, ǫ = 3.5kJ/mol), various degrees of hydrophobic repulsion between polar and nonpolar beads (V, ǫ = 3.1kJ/mol, VI, ǫ = 2.7kJ/mol, VII, ǫ = 2.3kJ/mol, and VIII, ǫ = 2.0kJ/mol), and the interaction between 20

21 charged particles and a very apolar medium (IX, ǫ = 2.0kJ/mol with σ = 0.62nm). For all ten interaction types the same effective size is assumed, σ = 0.47nm, except for the the level IX (σ = 0.62nm) and in special case, namely for ring particles. The ring particles (labeled S ) are introduced to model molecules containing rings. The mapping of atoms to ring molecules is 2 or 3 to 1. The effective minimum distance σ for the ring particles is set to 0.43nm and the strength of their interaction ǫ is scaled to 75% of the original value. In the simulations the LJ interaction potential is cut-off at a distance Ö ÙØ = 1.2 nm, which for most particle pairs corresponds to approximately 2.5σ. In addition to the LJ interaction, the model takes into account the electrostatic interactions between charged particles. In order to mimic screening, the Coulomb potential energy, Í Ð Ö µ Õ Õ πǫ ¼ ǫ Ö Ö (2 14) is shifted by adding a function S(r) such that the effective potential Í Ð Ö µ Ë Ö µ smoothly decays to zero as r approaches the cut-off radius of 1.2 nm. The relative dielectric constant ǫ Ö = 15 is also introduced to mimic the screening. Bonded interaction between chemically connected sites are described by a weak harmonic potential Í ÓÒ Ö µ with an equilibrium distance Ö ÓÒ = 0.47 nm for all particles except for ring particles (Ö ÓÒ = 0.43 nm) and particles used for the glycerol backbone in phospholipids (Ö ÓÒ = 0.37 nm), Í ÓÒ Ö µ ½ ¾ ÓÒ Ö Ö ÓÒ µ ¾ (2 15) where ÓÒ = 1250 kj/mol/nm ¾ is the force constant and Ö is the instantaneous bond length. The chain stiffness is modeled by a weak harmonic potential Í Ò Ð θµ for bond angles: Í Ò Ð θµ ½ ¾ Ò Ð Ó θµ Ó θ ¼ µ ¾ (2 16) where Ò Ð = 25 kj/mol/rad ¾ is the force constant and θ ¼ is the equilibrium bond angle, which depends on geometry of the molecule. 21

22 The cell membrane is modeled by a dipalmitoylphosphatidylcholine (DPPC) lipid bilayer. The DPPC molecule is comprised of the hydrophilic head group, which includes ester backbone, and two hydrophobic tails. The CG model for this molecule, shown in Fig. 2-1A, contains two charged beads (É) for the zwitterionic PC head group, two non-polar beads (Æ) for the glycerol ester backbone, and four apolar beads ( ) for each of the two tails. The CG model for CNTs is based on the model proposed by Wallace and Sansom (2007) [56]. The model CNTs are composed of the hydrophobic beads, similar to the beads used for the lipid tail-groups. Open-end zigzag (6,0) SWCNTs is considered (see Fig. 2-1C). The diameter of these nanotubes is 1.56 nm and length of their unit cells is 1.3 nm. In order to examine effects of the nanotube length, nanotubes containing three, four, and six unit cells were considered. In what follows, these nanotubes are denoted as CNT, CNT, and CNT, respectively. The model nanotubes are composed of hydrophobic beads whose LJ parameters are the same as those of the tail beads in a DPPC lipid. The bond and angle potentials are assumed to be harmonic, and the force constants for the bond and angle potentials were 2500 kj/mol/nm ¾ and kj/mol/rad ¾, respectively, chosen to make the CNTs rigid enough to maintain their shape against surrounding molecules. It was verified that the bond length and angle are normally distributed from the equilibrium length (0.47 nm) and angle (120 degree) within 0.06 nm and 2.1 degree, respectively, with 95% confidence level. An integration time step was reduced to 0.02 ps from 0.04 ps, which was used for simulations containing no CNTs, by considering the fastest frequency of bond vibrations. D Rozario et al. [5] is followed to approximate the structure of the fullerene C ¼ by 20 CG beads evenly spaced on a sphere of diameter 1.2nm. The force constants were the same as those for CNTs. The LJ parameters of these beads were taken to be the same as those for CG model of benzene [28]. In order to investigate effects of functionalization of carbon-based nanoparticles by hydrophilic groups, a CG model of a 22

23 fullerenol (functionalized fullerene), C ¼ (OH) ½¼ is considered. This model was obtained by replacing 10 hydrophobic beads on one of the hemispheres of C ¼ by 10 hydrophilic beads. The integration time step (0.02 ps) was used. A B Figure 2-1. CG models of (A) DPPC lipid molecule, (B) fullerene (left) and fullerenol (right), and (C) CNT. C A computational model for lipid bilayer is prepared through self-assembly. MD simulations of a DPPC lipid solution in water in a simulation cell of size nm were performed. The system contained 264 DPPC molecules and 4832 water beads. The initial conditions for this simulation were chosen to be a random dispersion of lipids in water (Fig. 2-2A). It was found that lipids self-assembled into a bilayer shown in Fig. 2-2B within a few nanoseconds. Temperature and pressure were maintained at 323 K and 1 bar. After a lipid bilayer is prepared, semiisotropic pressure coupling method is applied to maintain zero surface tension of the bilayer membrane for all considered systems. The box size changed to º½ º½ ½½º¼ nm from the self-assembly 23

24 simulations in order to ensure zero membrane tension γ and 1 bar pressure È Æ in the normal direction. All systems were rotated so that the bilayer is parallel to the x-y plane. A B Figure 2-2. Self-assembly of DPPC bilayer [50]: (A) initial random dispersion of DPPC beads in water and (B) self-assembled DPPC lipid bilayer. For clarity, the water molecules are not shown. 2.3 Stochastic Model and CMF Method Transport of even small molecules across a lipid bilayer takes place on a time scale that is out of reach of MD simulations. It is therefore not practical to study the transport effects of nanoparticles by direct MD simulations. Several techniques have been developed to enable MD simulations of such rare events [2, 15, 20]. In this investigation the Constrained Mean Force (CMF) method, which has been previously applied to similar systems[26, 27] was employed. The main assumption underlying this analysis is that nanoparticle transport can be described by the generalized Langevin equation [7], ÑĐÞ Øµ Ø ¼ γ Þ Ø τ µ Þ τ µ τ Þµ Þ Þ Øµº (2 17) Here, Þ is a reaction coordinate, Ñ is the mass of the nanoparticle, γ is the memory friction kernel, is the free energy and is the normally distributed random force with zero mean and the autocorrelation function obeying the fluctuation-dissipation theorem, 24

25 Þ Øµ ¼ (2 18) Þ Øµ Þ Ø τ µ γ Þ τ µ Ì º (2 19) The brackets denote the ensemble average, is the Boltzmann constant, and Ì is the system temperature. The fluctuation-dissipation theorem allows us to compute the friction coefficient from the random force autocorrelation function, which is readily measurable from MD simulations. If the correlation time for is very small then the friction with memory can be replaced by memoryless friction with coefficient γ. In this case, Eqs and 2 19 simplify to: Þµ ÑĐÞ Øµ γ Þµ Þ Øµ Þ Øµ (2 20) Þ Þ Øµ Þ Ø τ µ ¾γ Þµδ τ µ Ì (2 21) where δ is the Dirac delta-function. Further, the obtained friction coefficient is directly related to local diffusivity [7], Þµ Ì/γ Þµº (2 22) Let us now discuss the computation of the terms of Eq from the constraint MD simulations. If Þ is held constant by application of a constraint force,, then the first and second terms of Eq are zero. Further, the ensemble average of the random force term is zero, see Eq Therefore, the ensemble average of Eq yields the following relationship between the constraint force and the free energy: Þ ¼ ص Þ ¼µ º (2 23) Þ In analysis of the simulations the ensemble averaging is replaced by the time averaging using the ergodic assumption. 25

26 To fully reconstruct Eq it is necessary to obtain the friction coefficient, γ. To this end, Eq is applied to calculate the relationship between the time-independent friction coefficient and the autocorrelation function of the random force, ¾γ Þµδ τ µ Þ Øµ Þ Ø τ µ º (2 24) Ì Integrating both sides of the equation from 0 to infinity yields γ Þµ ¼ Þ Øµ Þ Ø τ µ غ (2 25) Ì The random force Þ Øµ is obtained by computing the deviation of the constrained force from its mean value, Þ Øµ Þ Øµ Þ Øµ º (2 26) 2.4 Calculation of Elastic Properties of Membrane Estimation of Bending and Tilt Modulus One of the possible effects of nanoparticles embedded in a membrane is a change of the membrane elastic properties, which in turn can lead to a membrane instability or a failure of some of the membrane functions. In this section, the relevant elastic properties of the membrane and methods of their calculation are briefly discussed. The classical membrane model was proposed by Helfrich [11]. This model assumes that the dominant contribution to the membrane energy is associated with the membrane bending. In this case, the membrane free energy can be approximated as À [ κ ¾   µ ¾ κã ] (2 27) where  and à are the mean and Gaussian curvatures of the membrane, respectively,  is the spontaneous curvature, κ is the bending modulus, κ is the Gaussian modulus, and the integration is performed over the membrane surface. This model approximates the membrane as an infinitely thin elastic sheet and neglects the membrane internal structure. 26

27 However, interaction of the membrane with nanoparticles is likely to cause localized disturbances in the neighborhood of the nanoparticles. Therefore, it is necessary to use the model which accounts for the internal membrane structure. Such a model was proposed by Hamm and Kozlov [10]. To account for the internal membrane structure, this model includes effects of lipid tilt. In order to quantitatively define tilt, a lipid director (Ò) is first defined as a vector pointing from the center of mass of the lipid tails to that of the head-group. Tilt Ø is then defined as the deviation of the lipid director Ò from the normal Æ to the surface of the monolayer containing this lipid [10], Ø Ò Æº (2 28) Ò Æ Here, vectors Æ and Ò are of unit length. The Hamm-Kozlov model approximates the bilayer energy as a sum of free energies of the monolayers comprising it. The monolayer free energy is approximated as [ ] ½ Àà ¾ κ   µ ¾ κ à ½ κ θø ¾ º (2 29) ¾ Here,  and à are the mean and Gaussian splays which contain contribution of both the membrane bending and the lipid tilt and κ θ is the tilt modulus. The integration over the Gaussian splay term yields the same result for all membranes containing no tears or discontinuities [30]. Therefore, contribution of this term to the membrane free energy can be neglected and the energy of a monolayer with vanishing spontaneous total curvature is [ Àà ½ ] κâ ¾ Ü Ý µ κ θ Ø ¾ Ü Ý µ Ü Ý º (2 30) ¾ It follows from the Eq and the equipartition theorem [30] that the magnitudes of equilibrium fluctuations of the monolayer surface Ü Ý µ and the molecular tilt Ø Ü Ý µ are 27

28 given by and Õµ ¾ Ì ( ½ κ Õ ½ κ θ Õ ¾ ) (2 31) ØÜ Õµ ¾ ØÝ Õµ ¾ Ì κ θ º (2 32) Here, Õµ and Ø Õµ are the Fourier transforms of the membrane surface Ü Ý µ and the lipid tilt Ø Ü Ý µ and Õ is the wave vector. The interface configuration of the membrane is evaluated by analyzing continuous approximation to the instantaneous dividing surface at each time-step. The dividing surface at time Ø is defined as a surface passing through the pivot points (x (t), y (t), z (t)), i.e. mid-points of bonds connecting lipid head- and tail-groups. In order to smooth the instantaneous dividing surface, the least-squares fit is used to obtain its Fourier series expansion, Ü Ý Øµ Õ Øµ ÕÜ Ü ÕÝ Ý µ (2 33) Õ Þ Øµ Ü Ý Øµ ¾ Ñ Òº (2 34) Normal vector Æ is obtained from the obtained instantaneous interface configuration necessary for calculation of the tilt vector (see Eq. 2 28), Æ ½ Ü Ý ½µº (2 35) ½ ¾ Ü Ý ¾ Here, Ü and Ý are the partial derivatives of Ü Ý µ with respect to Ü and Ý, respectively. Eq indicates that the magnitude of the long wavelength fluctuations of the monolayer surface (Õ 0) scales as Õ, whereas the short wavelength fluctuations scale as Õ ¾. The cross-over between these two regimes depends on the values of and θ and typically corresponds to the wavelength comparable with the membrane thickness (see section 4.4). 28

29 Eq enables us to estimate the bending and tilt modulus from simulations of a membrane at equilibrium. Specifically, and θ can be obtained by performing a least-squares fit of the magnitude of the bending and tilt fluctuations to Eq Note that θ influences to both the bending and tilt fluctuations, which allows us to check the self-consistency of the values of θ obtained from Eqs and Statistical Analysis of Correlated Time Series Theory As discussed in Section 4.4, relatively small differences between elasticities of the pure bilayers and bilayers containing nanoparticles are observed. It is therefore necessary to rule out a possibility that these differences can be attributed to statistical uncertainties. The bending modes most relevant in the calculation of the bending modulus correspond to long wavelengths ( ½¼ nm). The timescale of fluctuations of these modes is relatively slow, O(10 ¾ ) O(10 ) ns. Since the simulation timescale is typically O(½¼ ) ns, it is not practical to use only uncorrelated data in the analysis, since this will not provide us with sufficiently large sample for statistical analysis. It is necessary to account for correlation in this analysis, since error estimates obtained assuming zero correlation are not appropriate for data correlated in time (see Section ). Assume that each of the bending and tilt modes is independent of all other modes. Consider one of these modes and denote it by Þ Øµ to simplify the notation in this section. Assume furthermore that the dynamics of Þ Øµ can be described by a linear Langevin equation γ Þ Ø α Þ Þ ¼µ صº (2 36) Here, α R is the spring constant, γ R is the friction coefficient, and ص C is the Gaussian random force with zero mean. The random force and the friction coefficient 29

30 are related by the fluctuation-dissipation theorem ص µ ¾ Ìγδ δ Ø µ Ö ÓÖ Ñº (2 37) The latter assumption implies that [7] 1. Real and imaginary parts of Þ Øµ are independent of each other, ζ Ö ζ Ñ ¼ (2 38) where ζ ص Þ Øµ Þ ¼ is the deviation of Þ Øµ from its mean value Þ ¼ Þ and subscripts Ö and Ñ denote the real and imaginary parts of a complex variable. 2. Autocorrelation functions of ζ Ö Øµ and ζ Ñ Øµ are identical and decay exponentially, ζ Øµζ µ Ø /τ Ö ÓÖ Ñº (2 39) Assume that MD data are saved at times Ø Ò Ò ½µ Ø, Ò ½ ººº Æ. In order to connect these data with the theoretical model (2 36), a discretized version of this model is needed to obtain. For this, integrate Eq. (2 36) from Ø Ò ½ to Ø Ò, ζ Ø Ò µ φζ Ø Ò ½ µ ω Ò º (2 40) Here, ω Ò C are independent Gaussian variables with zero mean and variance σ ¾ ω. Parameters of the discrete equation (2 40) are related to parameters of the Langevin equation (2 36) by the following relationships: Moreover [46], φ Ø/τ τ γ α σ¾ ω Ì α ½ φ¾ µº (2 41) σ ¾ ζ ½ φ¾ µ ½ σ ¾ ω º (2 42) One of the approaches to obtain the parameters of the stochastic process (2 40) is the Yule-Walker method (see, e.g. Ref. [46]). This approach involves a least-squares fit to the autocorrelation function of ζ Ø Ò µ. However, this approach assumes that the 30

31 stochastic process has a zero mean and does not provide a simple way to estimate the error of Þ ¼ Since application of this method is anticipated to analyze shapes of deformed membranes, an alternative approach is chosen, which will allow to estimate Þ ¼ and its standard deviation. This is a maximum likelihood (ML) method [46], in which the vector of the system parameters Ü Þ ¼ Ö Þ ¼ Ñ φ σω ¾ µ is obtained by maximizing the likelihood function Ä Ü Þµ ÐÒ È Þ Üµ Æ ÐÒ σ ¾ ω ÐÒ ½ φ¾ µ Ë ½ σ ¾ ω Ñ Ü (2 43) for a given realization of time series Þ Þ ½ ººº Þ Æ µ. Here, Þ Ò Þ Ø Ò µ, È Þ Üµ is the probability to observe the realization Þ of the time series if the values of the system parameters are Ü, and Æ Ë ½ ½ φ ¾ µ Þ ½ ¾ Þ Ò φþ Ò ½ ¾ º (2 44) Note that in the case of uncorrelated time series (i.e., φ ¼) and an a priori known σω, ¾ maximization of (2 43) corresponds to solution of the usual least squares problem, Ò ¾ Ä ½ σ ¾ ω Æ Þ Ò ¾ Ñ Ü º (2 45) Ò ½ Numerical algorithm It was found that the standard optimization algorithms (such as the methods of conjugate gradients and steepest descent) do not always converge or converge very slowly to the maximum of the likelihood function Ä. The problem is associated with the stiffness of this function, i.e. a very non-uniform dependence of Ä on its parameters. To see this, consider Ä σω ¾ µ at fixed Þ ¼ and φ, Ä σ ¾ ω µ Æ ÐÒ σ¾ ω Ë ½ σ ¾ ω ÓÒ Øº (2 46) 31

32 When σω ¾, the function Ä σω ¾ µ is dominated by the first term of Eq. (2 46), i.e. this function varies slowly when σ ¾ ω is sufficiently large. On the other hand, when σ ¾ ω ¼, the function Ä σω ¾ µ is dominated by the quickly varying second term of Eq. (2 46). This implies that an iteration of a standard optimization method is likely to overshoot when moving from the region of the slow variation of Ä σω ¾ µ to the region of the fast variation. Therefore, an alternative numerical scheme which takes advantage of the specific structure (2 43) of the likelihood function is developed. A critical point of Ä corresponds to the solution of the following system of equations where Ë ½ is defined by Eq. (2 44), [ scheme: Ë ¾ Ë Ä φ ¾φ ½ φ ¾ Ë ¾ σω ¾ φë µ ¼ (2 47) Ä Æ Ë½ ¼ σω ¾ σω ¾ σω (2 48) Ä ¾ Ë ¾ Þ ¼ σω ¾ (2 49) ½ φµ ζ ½ ζ Ò µ (½ φ Æ ½ ζ Ò ¾ Ë Ò ¾ ] ζ Ò (2 50) ¾) Æ ½ Ò ¾ Æ Ê ζ Ò ½ ζ Ò µ Ò ζ Ò Þ Ò Þ ¼ º (2 51) Ò ¾ The system of equations (2 47)-(2 49) is solved using the following iteration 1. Initial guesses for Þ ¼ and σω ¾ are taken to be the values of the mean and standard deviations of Þ Ò obtained assuming the zero correlation time, i.e. Þ ¼µ ¼ ½ Æ Æ Ò ½ Þ Ò [ σ ¾ ω ] ¼µ ½ Æ ½ Æ Ò ½ Þ Ò Þ ¼µ ¼ µ¾ º (2 52) 2. Solve equation (2 47) for φ at fixed σ ¾ ω and Þ ¼. This equation is solved by Newton s method with the initial guess φ Ë /Ë. 3. Solve equation (2 48) for σ ¾ ω, σ ¾ ω Ë ½ Æ º (2 53) 32

33 4. Solve equation (2 49) for Þ ¼, Þ ¼ Ë ¾ Æ ½ φµ ¾φ (2 54) at fixed φ and σω ¾. 5. Repeat steps 2-4 until convergence. Steps 2, 3, and 4 yield a unique maximum of Ä as a function of φ, σω ¾, and Þ ¼, respectively, with other variables held constant, since ¾ Ä φ ¾ < ¼ ¾ Ä σ ¾ ω µ¾ < ¼ Ò ¾ Ä Þ ¾ ¼ < ¼ Ö Ñ ÓÖ ÐÐ φ ¼ ½µº (2 55) Therefore, this algorithm converges to the maximum of the likelihood function and this maximum is unique for φ ¼ ½µ Error estimation Let Ü be a solution of (2 43) for a given time series Þ, i.e. Ä Ü Þµ ¼ and Ä Ü Þµ is a negative-definite matrix. The probability distribution of the system parameters Ü is È Ü Þµ Ä Ü Þµ ÜÔ [ ] ½ ܵ Ä Üµ Ü Üµ ÀºÇºÌ º (2 56) ¾ Ü Neglecting the higher order terms (H.O.T.) in (2 56), it is obtained that the maximum likelihood solution is unbiased, i.e. Ü Ü, and the covariance matrix of the model parameters is Ü Üµ Ü Üµ Ä Üµ ½ º (2 57) Generalizing the derivation of Ref. [12] to the case of complex time series Þ, Ð Ñ Æ ½ Æ Ä Ð Ñ Æ ½ Æ Ä σ ¾ Þ ¼ ¼ ¼ ¼ ¼ σ ¾ Þ ¼ ¼ ¼ ¼ ¼ σ ¾ φ ¼ ¼ ¼ σ ω ¼ (2 58) 33

34 is obtained. Here, σ ¾ Þ ¼ σ ¾ φ σ ¾ ω ¾ ½ φµ ½ φµ ¾φ/Æ (2 59) ½ φ ¾ µ ¾ ¾ ½ φ ¾ µ φ ¾ ½µ/Æ (2 60) Combine (2 57) with (2 58) to obtain Ð Ñ Æ ½ Æ σ ¾ Þ ¼ ¼ ¼ ¼ ¼ σ ¾ Þ ¼ ¼ ¼ ¼ ¼ σφ ¾ ¼ ¼ ¼ ¼ σω º (2 61) Therefore, error estimates for Þ ¼ Ö Ñµ, φ, and σ ¾ ω are σ Þ¼ / Æ σ φ / Æ Ò σ ¾ ω / Æ (2 62) respectively Validation In order to validate the Maximum Likelihood method and to demonstrate its necessity for analysis of the correlated data, a series of simulations of the Langevin equation (2 36) for different sets of parameters corresponding to the correlation times τ/ Ø ½, 10, 40, and 100 is obtained. The simulations for the same set of parameters Ü a number of times were repeated in order to obtain a distribution of the estimated parameters Ü. The standard deviation of the estimated system parameters computed from this distribution represents the true error of this estimation. Of course, in reality it is necessary to obtain the error estimates based on a single realization of the stochastic process (2 36). The latter estimates are the estimates (2 62) provided by the ML method. It was found that for all considered parameters of the Langevin equation, the ML method yields an accurate estimate of the system parameters and the error 34

35 estimates obtained by the ML methods are in good agreement with the true errors. On the other hand, error estimates based on the assumption of uncorrelated time series underestimate the true errors by an order of magnitude for τ ¼ Ø ½¼¼ Ø and a factor of two for τ ½¼ Ø Stress Tensor In addition to changes in the membrane stability, the embedded nanoparticles may affect the membrane function by changing activity of membrane proteins. In particular, some ion channels are sensitive to the lateral pressure profile in the membrane [4]. Therefore, nanoparticle-induced changes in the lateral pressure may shift the equilibrium balance between open and closed states of the channels. In order to study spatial variation of the pressure tensor È within the membrane, instantaneous local stress tensor σ, È σ (2 63) is averaged. The stress tensor at the microscopic level consists of two parts, namely the kinetic part due to momentum transport and a potential part arising from intermolecular forces [45], σ αβ Ê Øµ ½ Î Ñ Ú α Ú β Ö α Í {Ö }µ ¼ Ð β δ Ê Ðµº (2 64) Here, α β Ü Ý or Þ, Î is the volume, Ñ, Ö, and Ú are the mass, velocity, and position of the -th particle, ¼ is a contour connecting an arbitrary point Ê ¼ with the -th, and Í is the total potential energy, which can be expressed as Í {Ö }µ Í µ Ö ½ ººº Ö µº (2 65) < > 35

36 Here, Í µ are -body potentials and < > ½ ººº µ are particle indices. In this system, ¾ corresponds to Lennard-Jones, electrostatic, and bond length potentials and corresponds to the bond angle potential. As demonstrated in Ref. [8], Eqs and 2 65 yield the following expression for average of the instantaneous stress tensor over some domain Å: σ αβ Šص ½ Î Å Å σ αβ Ê Øµ Ê (2 66) ( ) Ñ Ú α Ú β ½ Í µ µ Í Î Å ÑÎ Å Ö α Ö α Ö β Ð Å Ö Ö Ð (2 67) µº Ð ½ Ö Å < > < Ð> Here, Ö Ð Ö Ð Ö is the vector pointing from atom to atom Ð and Å Ö Ö Ð µ Ö Å Ö Ö Ð µ Ö Ð ¼ ½ (2 68) is a weight function which specifies the contribution of interactions between atoms and Ð to the stress tensor in domain Å. Ö Å Ö Ö Ð µ is the segment of Ö Ð contained in domain Å. The integration contour Ð in Eq is an arbitrary contour connecting atoms and Ð and Å Ö Ö Ð µ is the fraction of this contour contained in domain Å. In this work, the Irving-Kirkwood (IK) contour [16] defined as a straight line connecting atoms and Ð, Ê λµ Ö Ð λö Ð λ ¼ ½ is used. In this case, assuming that the interval λ λ Ò Å Ö Ö Ð µ λ ÓÙØ Å Ö Ö Ð µ corresponds to the segment of the contour Ð inside the domain Å, the weight function can be expressed as Å Ö Ö Ð µ λ ÓÙØ Å Ö Ö Ð µ λ Ò Å Ö Ö Ð µ º (2 69) In the current work, one- and two-dimensional pressure profiles are computed by averaging the stress tensor over the following domains (see Fig. 2-3): 1. Slabs of thickness Þ parallel to the bilayer plane 2. Rectangular columns normal to the Ü Ý plane. Each edge of these columns is parallel to one of the axes of coordinates and the column cross-section is Ü Ý. 36

37 A B Figure 2-3. Cross-sections of (A) slabs and (B) columns in which the stress tensor is averaged. An IK contour connecting atoms and Ð is also shown. Contribution of the interaction between these atoms to a particular domain (shown by light grey) corresponds to the fraction of the contour contained inside the domain (shown by a thicker line). When computing the two-dimensional pressure profile of a system containing an anisotropic nanoparticle (such as a carbon nanotube), the system of coordinates is rotated so that the nanoparticle points along the Ü-axis. This change of coordinates is performed at each timestep after the stress tensor is computed. Therefore, it is necessary to reflect the coordinate change in the tensor components. In general, if new and old coordinate systems are related by a linear orthogonal transformation Ö Ò Û ÌÖ ÓÐ (2 70) the corresponding stress tensors are related by σ Ò Û Ìσ ÓÐ Ì ½ º (2 71) 37

38 In the current case, the transformation Ì is the rotation around the Þ axis by angle φ between the nanoparticle direction and the original Ü-axis, i.e. Ó φ Ò φ ¼ Ì Ò φ Ó φ ¼ º (2 72) ¼ ¼ ½ 38

39 CHAPTER 3 TRANSPORT OF CARBON-BASED NANOPARTICLES THROUGH LIPID MEMBRANES 3.1 Introduction As discussed in the previous chapter, the small size of nanoparticles makes them promising candidates for new generation of drug delivery vehicles and diagnostic tools. However, possibility of harmful effects of nanoparticles on cellular membranes is of increasing concern. Some experiments have shown toxicity of nanoparticles such as death of cultured cells exposed to multi-wall carbon nanotubes, carbon nanofibers, and carbon black [25]. Although exact mechanisms of cytotoxicity are not clear, it has been shown that cell death takes place after the nanoparticles contact with cell membranes or after they are internalized by cell. Rothen-Rutishauser et al. [42] have shown that polystyrene, gold and TiO ¾ nanoparticles can enter red blood cells by an unknown mechanism different from endocytosis (a common mechanism of internalization of molecules, which involves interaction of the molecules with specific binding proteins in the cell membrane). Therefore, understanding nanoparticle toxicity requires understanding of the mechanisms of nanoparticle permeation into cytoplasm. In addition to experments, molecular modeling of the interactions of nanoparticles with cell membrane may provide insight into toxicity mechanisms of nanoparticles. Computer simulations have been used to explore the interactions between nanoparticles and biological membranes. In particular, MD simulation is a useful tool for investigation of the permeation of nanoparticles into lipid membranes. They provide spatial and temporal resolution inaccessible by experiments. Molecular dynamics studies have shown translocation of C ¼ fullerene and its derivatives through lipid membranes. Fullerene aggregates formed in water penetrated into a lipid membrane and stayed as monomeric fullerene in the membrane interior [59]. On the other hand, fullerene derivatives absorbed to the bilayer surface and remained at the bilayer-water interface [5, 41], which was said to make fullerene derivatives less toxic than pristine fullerenes. 39

40 In addition to the nanoparticle partitions in lipid bilayers, efforts have been made in examining changes of membrane properties caused by the particles. MD simulations of Wong-Ekkabut et al. [59] showed that high fullerene concentration in a lipid membrane affects its elastic and structural properties. Fullerenes induced an increase in both the projected area and the thickness of the bilayer but decrease in the bending modulus. In addition, fullerene and some fullerene derivatives have been shown to experience a small energy barrier for their entry into a lipid bilayer [5]. In this study, it is demonstrated that not only the nanoparticles may cause the membrane deformation but that the deformation itself plays a significant role in the nanoparticle transport. In the current chapter, interactions between fullerenes and a lipid membrane during their transport across the lipid membrane are evaluated. The permeation process into a flat lipid bilayer is considered neglecting the curvature effects of a cell. This is a good approximation since the typical cell size is much larger than the nanoparticles of interest. In addition, lipid bilayers are simulated by neglecting the presence of the membrane proteins based on the experimental evidence that nanoparticle transport may occur by a process different from endocytosis. Similar membranes without the membrane proteins were used in research of transport of carbon-based nanoparticles [51] and fatty acids across membranes [53]. 3.2 Free Energy Profiles To put the obtained results in perspective, in Fig. 3-1A the density profile of the lipid bilayer is shown. The free energy profiles obtained for the considered nanoparticles are shown in Fig. 3-1B. Recall that the system of coordinates is oriented so that the Þ-axis points along the bilayer normal and the bilayer center of mass is located at Þ ¼ nm. As expected, the fullerene experiences a significant decrease of the free energy in the bilayer interior, i.e. for -2 nm < Þ < 2 nm. This preference of fullerenes to the bilayer interior is caused by the stronger attraction of hydrophobic nanoparticles to the hydrophobic DPPC tails, as compared to the hydrophilic head-groups or 40

41 water molecules. These strong attractive interactions provide a driving force for the nanoparticle to remain in the bilayer interior for a very long time. The shift of the energy minima of C ¼ occurs in the soft polymer region of the lipid bilayer [27] characterized by the high density of ordered tail groups. As can be seen from Fig. 3-1B, this region corresponds to 0.5 nm < Þ < 1.0 nm for C ¼. The nanoparticle is likely to be located at the bilayer center due to the free volume caused by the decreased density of the tail region at 0 nm < Þ < 0.5 nm (see Fig. 3-1A). However, strong atractive van der Waals interaction between the particle and the lipid tails in the highest density region (0.5 nm < Þ < 1.0 nm) appears to pull the particle slightly away from the point of the lowest tail density. In contrast, the free energy profile of C ¼ (OH) ½¼ exhibits the energy minima at 1.5 nm < Þ < 2.0 nm. As shown in Fig. 3-1A, lipid head-groups are present at 1.5 nm < Þ < 2.0 nm. They attract the functional groups of the particle toward the water phase. The repulsion between the hydrophilic groups of the particle and tail-groups induces the high energy barrier at the bilayer center. The preferential location of hydrophobic C ¼ and amphiphilic C ¼ (OH) ½¼ are in the neighborhood of the membrane center and at the boundary between the head- and tail-groups of the membrane, respectively. Both C ¼ and C ¼ (OH) ½¼ experience relatively small energy barrier ( 4 kj/mol) to enter the membrane. This can be explained by the following observation. The strength of nanoparticle interaction with hydrophilic lipid head-group is very similar to that of the interaction with water. Further, the interactions of the hydrophobic nanoparticle with the glycerol ester backbone are slightly more attractive and the interactions with lipid tail beads are significantly more attractive than the interactions with water. Of course, the energy required to move apart the lipid head-groups that are chemically bonded with relatively bulky tail-groups is greater than the energy required to move apart small water molecules. However, as the density profile in Fig. 3-1A illustrates, the bilayer interface contains a large proportion of water beads, which mitigates the need, at least in this 41

42 10 60 C 60 Density (beads/nm 3 ) Total Water Head Group Tail Group G (kj/mol) C 60 (OH) z (nm) A z (nm) B Figure 3-1. (A) Density profile of the DPPC bilayer. The solid line shows the average number of beads per cubic nanometer, while the remaining lines indicate the density of water, head group and tail group beads. (B) Free energy profiles for fullerene (solid line) and fullerenol (dashed-dotted line). region, to move the DPPC beads apart. Once the nanoparticle is within the range of the glycerol ester backbone, the drive towards more favorable interactions becomes significant. This drive is further magnified as the nanoparticle approaches the tail region. x D (m 2 /s) z (nm) Figure 3-2. Dependence of the diffusivity of fullerene on the nanoparticle position within the bilayer. As can be seen in Fig. 3-2, the nanoparticle diffusivity is highly inhomogeneous inside the bilayer. Moreover, all of the considered nanoparticles experience a sharp decrease of diffusivity near the entry point into the membrane. It has been recently 42

43 demonstrated [9] that such decrease characterizes strong coupling between the nanoparticle transport and the membrane undulation. In particular, the membrane shape is expected to be significantly perturbed. Such membrane deformation is indeed observed when a nanoparticle is located near the membrane entry points. In Fig. 3-3 average dividing surfaces of the upper and lower membrane leaflets are shown when the nanoparticle is located in the lower leaflet and the nanoparticle location corresponds to the minimum of the diffusion coefficient D. The lower leaflet sharply protrudes toward the nanoparticle in its direct neighborhood. This protrusion is followed by a slow decay towards the equilibrium state as the distance from the nanoparticle increases. The upper leaflet deforms toward the bilayer center around the fullerene and slowly returns to the equilibrium state as the distance from the nanoparticle increases. The hydrophobic groups of C ¼ and C ¼ (OH) ½¼ attract the hydrophobic tails of lipids, which results in the deformation of the lower leaflet towards the nanoparticle. This, in turn pulls the upper leaflet in the same direction. 4 3 C 60 C 60 (OH) 10 2 < h > (nm) r (nm) Figure 3-3. Local deformation of a lipid membrane containing a fullerene or a fullerenol: Average dividing surfaces of the top and bottom leaflets of the membrane when the fullerene (solid line, black circle) is constrained at Þ = -3.1 nm and the fullerenol (dashed line, hollow circle) is constrained at Þ = -2.7 nm; Ö denotes the distance from the nanoparticle center of mass. Fig. 3-4 shows the dividing surfaces of the bilayer corresponding to several nanoparticle positions. When the nanoparticle is away from the bilayer, the average 43

44 surfaces of bilayer are flat for both types of fullerenes. However, the membrane deformation is extremely sensitive to the nanoparticle position as it approaches the membrane, This can be seen further from Fig. 3-5, which shows the dependence of one of the Fourier modes in the expansion 2 33 of the dividing surface on the nanoparticle position. The leaflets containing C ¼ and C ¼ (OH) ½¼ undergo the large shape changes at Þ = 3.1 nm and Þ = 2.7 nm, respectively. There is a steep increase of the membrane deformation at the membrane entry points corresponding to small diffusion coefficient D. This implies that a small change in the nanoparticle position causes substantial changes in the membrane undulations. 6 4 z = 4.0 nm z = 2.7 nm z = 0.0 nm < h > (nm) r (nm) Figure 3-4. Average dividing surfaces corresponding to the fullerenol constrained at Þ = -4.0 nm (dash dotted line, gray circle), Þ = -2.7 nm (dashed line, hollow circle) and Þ = 0.0 nm (solid line, black circle). Such sharp changes lead to a strong coupling between membrane fluctuations and nanoparticle motion [9]. The coupling is caused by an attractive interaction between lipid tails and hydrophobic groups of the particles. Since the time scale of the deformation is much longer than that of the nanoparticle diffusion, the deformation leads to a long correlation time of the random force acting on the particle. Indeed, the force auto correlation function (ACF) exhibits a slow decay when the nanoparticle is constrained near the points corresponding to strong membrane deformation as shown Fig For all nanoparticle positions, the force ACF exhibits fast initial decay. For positions 44

45 Re <h> (nm) C 60 C (OH) z (nm) Figure 3-5. Dependence of the Fourier mode with wavenumber Õ = 0.46 nm ½ of the dividing surface of the leaflet containing the nanoparticle on the nanoparticle position Þ. inside the membrane, the ACF also exhibits fast oscillations. These oscillations and fast decay occur due to interactions between individual atoms. The interactions of the slow collective membrane degrees of freedom with the nanoparticle correspond to the slow decay of ACF, which is observed only for specific locations inside the membrane. In order to estimate the rate of the slow decay, the envelop of the ACF to a double-exponential function is fitted. The envelop is obtained by tracing local maxima and minima of the ACF, as illustrated in the inset of Fig Dependence of the obtained correlation time on the fullerene position is shown in Fig The correlation time of the random force is fairly small at most nanoparticle locations but is rather large at the entry points into the bilayer, which correspond to the sharp changes of the membrane deformation (see Fig. 3-7). C ¼ and C ¼ (OH) ½¼ have different effects on the membrane deformation as they move further towards the membrane center. The bilayer containing C ¼ returns back to the undeformed flat state when the nanoparticle is located at the bilayer center. On the other hand, C ¼ (OH) ½¼ increases the membrane deformation as it moves closer to the membrane center (see Fig. 3-5). This deformation is caused by the hydrophilic functional groups, -OH, of C ¼ (OH) ½¼, which attract the bilayer head groups. 45

46 10 0 C(τ) 10 1 C(τ) τ (ps) 10 2 (b) (a) (c) τ (ps) Figure 3-6. Envelops of ACF of the random force acting on the fullerenol constrained at (A) z = -4.0 nm (dashed-dotted line), (B) z = -2.7 nm (dashed line), and (C) z = 0.0 nm (solid line). In this plot ACFs are normalized so that C(τ)=1. The inset shows C(τ) (solid line) and fitted envelops (dashed line) for small τ C 60 C 60 (OH) 10 τ (ps) z (nm) Figure 3-7. Correlation times of the slowest fluctuations of the random force acting on particles. 3.3 Effect of Fullerenol Orientation on Membrane Deformation In this section, an effect of orientation of fullerenol C ¼ (OH) ½¼ on the membrane deformation and free energy profile is considered. The orientation of the amphiphilic fullerenol is defined as the cosine of angle θ between the z-axis and a fullerenol director. The latter is a vector pointing from the center of mass of the hydrophilic groups to that of the hydrophobic groups as shown in Fig. 3-8A. The z-axis is oriented so that the z coordinate of the particle center of mass is positive. Then cosθ = 1 indicates that the 46

47 hydrophilic part of C ¼ (OH) ½¼ points downward and the plane separating the hydrophilic and hydrophobic part is exactly parallel to the bilayer surface. In order to obtain a relationship between C ¼ (OH) ½¼ position and orientation inside bilayer, the contribution of of the orientation to the free energy Ó θ Þµ is computed. Ó θ Þµ is obtained from MD simulations with constrained z. These simulations allow us to compute the probability È Ó θ Þµ of the nanoparticle orientation when it is constrained at position Þ. Since the system obeys the Boltzmann distribution, the following contribution of the nanoparticle orientation to the system free energy Ó θ Þµ is considered: Ó θ Þµ ÌÐÒÈ Ó θ Þµº (3 1) A B Figure 3-8. (A) Definition of the fullerenol orientation: angle between C ¼ (OH) ½¼ director (d) and the z-axis. Gray and white spheres indicate hydrophilic and hydrophobic beads of fullerenol, respectively. (B) Contribution of the fullerenol orientation to the free energy (kj/mol). Fig. 3-8B shows the obtained two-dimensional profile of Ó θ Þµ. As expected, the particle orientation is uniformly distributed away from the bilayer. As the particle approaches the bilayer, the functional group becomes oriented toward the hydrophilic head groups due to attraction between the functional groups of the particle and the lipid head-groups. The particle orientation remains in a narrow range as the nanoparticle 47

48 enters the membrane. The range of orientations becomes wider near the bilayer center. This trend is explained by a stronger attraction between the hydrophilic groups of the particle and lipid head-groups at the entry region than at the bilayer center. It was found that the minimum energy path obtained from the the energy profile is so noisy that a correct path cannot be determined followed by the nanoparticle. Therefore, a possible energy path is approximated, referred to as the dashed arrow, followed by the particle when it crosses the bilayer. Assume that a particle starts to move from a point of z = -6 nm and cosθ = 1. The particle prefers to maintain orientation of cosθ = 1 as it moves towards the bilayer center, but it changes its orientation to cosθ = -1 when it passes through the center and maintains this orientation until it leaves the bilayer at z = 3 nm. It is expected that the nanoparticle would follow a smooth transition of orientation in reality rather than the sharp change in the dashed line during its transprot. The dependence of the nanoparticle behavior on its position and orientation can be better understood by considering a 3-dimensional free energy profile schematically shown in Fig The schematic profile adds the observed orientation indicated as cosθ to the free energy profile shown in Fig. 3-1B. When a particle enters the bilayer with orientation of cosθ = 1, it follows the free energy profile on the front plane (1) at cosθ = 1. The particle has orientation of the cartoon on the plane (a) at the minimum free energy and maintains the orientation as it approaches the bilayer center. As the particle moves toward the other monolayer passing through the center marked as (*), it experiences an energy barrier higher than is evident from the 1-dimensional profile in Fig. 3-1B. This is explained by the necessity of the change of orientation (not captured by 1-dimensional profile). Since the change of the particle orientation from plane (1) to (2) takes place continuously, the particle follows the thicker dotted line in real transport rather than passing through a unrealistic sharp point observed in Fig. 3-1B. In what follows, a relationship between the fullerenol orientation and the interface shape is analyzed. To this end, both the position and orientation of the nanoparticle are 48

49 Figure 3-9. A schematic of free energy profile for fullerenol and its orientational behavior. The dark part of a sphere indicates a hydrophilic part of fullerenol. constrained. The distance between the center of mass of the fullerenol and the bilayer in the direction perpendicular to the bilayer surface and the nanoparticle orientation is constrained. The latter is constrained by keeping a constant distance between the center of mass of the fullerenol hydrophilic beads and the bilayer center. A detailed investigation was performed for the nanoparticle constrained at the bilayer center. In Fig the average height of dividing surfaces of the bilayer containing a fullerenol located at the bilayer center for the considered orientations, 0 and 90 degrees, is shown. As can be seen from Fig. 3-10A, the upper layer is significantly deformed when the functional group of the particle is pointing upward. The upper layer is attracted toward the hydrophilic part of the fullerenol. Deformation of this layer in turn causes the deformation of the lower layer. 49

50 A B Figure Average height of dividing surfaces of the bilayer containing a fullerenol located at the center and oriented at (A) 0 degree and (B) 90 degrees with respect to z-axis. A different membrane deformation is observed when Ó θ ¼, i.e. the plane separating the hydrophilic and hydrophobic part of the particle is perpendicular to the bilayer surface. As shown in Fig. 3-10B, this particle orientation leads to an asymmetric bilayer deformation. While the functional group attracts neighboring lipid head groups close to it, the head-groups of other lipids in the same leaflet but on the opposite side of the nanoparticle are out of the attraction range of the functional group. The unbalanced interaction with lipids on each side from the boundary results in the asymmetric deformed membrane. The membrane deformation was also considered for other particle orientations, namely 30, 45, and 60 degrees. These deformations show the transition between the membrane shapes observed at the 0 degree to 90 degree nanoparticle orientations. As the particle orientation increases from 0 to 90 degrees, the asymmetry with respect to the plane normal to the bilayer surface also increases, whereas the asymmetry with respect to the bilayer plane decreases. The asymmetry with respect to the vertical plane to the bilayer of the deformation is compared by computing a difference of the average height of the membrane surface for various nanoparticle orientations. between two 50

51 divided regions from the fullerenol center along x direction. The local average heights of lipids are computed in each region (region 1 and 2) defined in Fig. 3-11A. x = 0 corresponds to the center of the particle. The heights are averaged over the region 1 (-2 nm < Ü < 0 nm) and 2 (0 nm < Ü < 2 nm) along -1 nm < Ý < 1 nm. Fig. 3-11B shows dependence of the obtained average surface height on the particle orientation cosθ. It can be seen that difference in the average heights in the between two regions decreases as cosθ changes from 0 to 1. This indicates the asymmetry in the deformation from the boundary plane of the fullerenol. In summary, C ¼ (OH) ½¼ orientation plays an important role in the membrane deformation due to their interactions between the particle and membrane. 2 2nm < x < 0nm 0nm < x < 2nm Height (nm) A cosθ B Figure Comparison of local average heights of lipids around nanoparticles. (A) Top view of the monolayer surface. The white and gray boxes refer to region 1 and 2, respectively. (B) Dependence of local average heights of the upper monolayer surface in the region 1 and 2. The white and black bars in (B) correspond to the region 1 and 2, respectively. 3.4 Local Membrane Energy and Effective Interaction Potential between Nanoparticles Membrane-mediated interactions between the membrane inclusions (such as nanoparticles and proteins) occurs through mechanical stress created by membrane deformations. For example, two membrane inclusions would be attracted to each other if 51

52 decrease of the distance between them leads to a decrease in the total stress within the membrane. In the current section, the membrane deformation and the corresponding energy penalty caused by nanoparticles located at their preferred positions within the membrane are assessed. The membrane energy is estimated using the Hamm-Kozlov (HK) model [10]. The HK model describes elastic membrane deformation by accounting for energies of bending and lipid tail tilt. To apply this model, the mean dividing surface h(x,y) and lipid tilt t(x,y) defined in chapter are obtained in each of the bilayer leaflets. The lipid director is defined as a vector pointing from the center of mass of tails to that of heads. The system of coordinates used in these calculations was chosen so that (i) the origin coincides with the projection of the nanoparticle center of mass onto the x-y plane and (ii) the Ü-axis coincides with a director pointing from a center of hydrophilic groups to that of hydrophobic groups. As can be seen from Fig. 3-12A, the fullerenol creates a membrane protrusion around the particle. In addition, a relatively large deviation of lipid tilt is observed around the hydrophilic part of the particle from its bulk (zero) value, see Fig. 3-12B. Since lipids on the hydrophilic side are disrupted due to the hydrophilic attraction, lipid tails are likely to be tilted to avoid a repulsion between the tails and the hydrophilic part of the particle. Parameters in the HK model are obtained using the method proposed by May et al [30]. The bending and tilt moduli are estimated from the spectral intensity of membrane fluctuations (see Eq. 2 31) a least-squares fit. The corresponding membrane energy is shown in Fig The perturbation of the membrane energy is relatively small and is localized around the nanoparticle. Therefore, interactions between the nanoparticle and other membrane inclusions are rather weak and short-ranged. The short-ranged interaction is not likely to cause aggregation of the nanoparticles, which is consistent with results shown in other MD simulations [23, 59]. 52

53 A B Figure Effect of fullerenol embedded in a DPPC membrane on (A) upper and (B) lower membrane leaflet. See Fig. 4-8 for details. Figure Energy (kj/mol) of membrane containing a fullerenol. 3.5 Conclusions The calculated free energy profiles demonstrate that there is no significant energy barrier to enter the bilayer for any of the nanoparticles studied. This suggests that these nanoparticles may enter the bilayer relatively quickly. On the other hand, nanoparticles spend long time inside the bilayer due to a deep energy well in the bilayer interior. Therefore, the hydrophobic interior of a lipid bilayer acts as a trap for hydrophobic 53

54 particles. Since the considered nanoparticles will spend a significant amount of time within the cell membrane, they will have the opportunity to impact the membrane interior. It was observed that there is a strong coupling between the considered nanoparticle transport and the membrane undulation. The membrane was significantly perturbed near the entry points of the nanoparticle into the membrane, which implies that the nanoparticles may damage the membrane during their permeation into the membrane. Effects of orientation of the amphiphilic nanoparticle on the membrane perturbation were examined. The membrane deformation was affected by the orientation of the nanoparticle inside the bilayer due to the hydrophilic attraction between functionalized groups of the nanoparticle and lipid head-groups. Moreover, consideration of the orientation allows us to better understand the transport mechanism of the nanoparticle across the bilayer, since the transport involves particle reorientation. As predicted by the orientation-dependent free energy profile, the nanoparticle is expected to experience a higher energy barrier than that predicted by the free energy profile depending on position only. The amphiphilic particle induces a small and localized perturbation of the membrane energy when it is located at its equilibrium position. This implies that the range of interaction between nanoparticles is short. The weak and short-ranged interactions between the particles are unlikely to cause particle aggregation inside the membrane. 54

55 CHAPTER 4 ASSESSMENT OF POSSIBLE NEGATIVE EFFECTS OF CNTS ON LIPID MEMBRANES 4.1 Introduction As it was discussed in the previous chapter, the small size of nanoparticles enables them to easily penetrate cell membranes. Once inside a membrane the nanoparticles may spend long time interacting with its components. A number of experiments demonstrated changes in membrane structure induced by nanoparticles. E.g., membranes of human epidermal keratinocytes cells have been shown to undergo morphological changes upon exposure to SWCNTs [47]. Furthermore, some nanoparticles have been shown to cause breaks in cell membranes [38]. One of the proposed mechanisms of the observed membrane instability is lipid peroxidation caused by chemical reactions between nanoparticle-induced ROSs and lipids. In addition, recent studies have shown an alternative mechanism due to a non-reactive physical interaction between nanoparticles and membranes. For example, positively charged generation 7 polyamidoamine (PAMAM) dendrimers were shown to create nanoscale holes in dimyristoylphosphatidylcholine (DMPC) bilayer due to a physical interaction of the nanoparticles with the membrane [31]. Amine-terminated groups were attached to PAMAM dendrimers and electrostatic interaction between the positive charges on the dendrimer surface and lipids caused formation of the holes. On the other hand, electrically neutral generation 5 PAMAM dendrimers did not create any holes [13], thereby indicating that modification of a nanoparticle surface plays an important role in the interaction between nanoparticles and cells. Moreover, study of polycationic nanoparticles demonstrated that both nanoparticle size and charge contributed to the degree of disruption [22]. MD simulations have shown that carbon-based nanoparticles, such as fullerenes and some of fullerene derivatives experience a relatively small energy barrier to enter the lipid bilayer and their energy decreases significantly once they are inside the bilayer 55

56 [5, 41, 59]. This implies that these nanoparticles will easily permeate into the membrane interior and will reside inside the membrane for a long time. During their residence, nanoparticles may significantly perturb the membrane and even disrupt the membrane integrity. Possible sources of instability include lipid-mediated nanoparticle aggregation, changes of the membrane elastic properties, and formation of non-bilayer phases Recent MD investigations [59] of the changes of membrane properties due to embedded nanoparticles (fullerenes) report that the membrane lipids stretch in the neighborhood of the nanoparticle, which leads to the decrease of the membrane area per lipid. Moreover, it was also observed that the fullerenes affect elastic properties of the membrane. However, the latter changes are observed only at very high fullerene concentrations. In this chapter the physcial mechanisms of nanoparticle toxicity are focused on, i.e. it is assumed that no reaction takes place and the damage to the membrane is caused by such effects as changes of membrane elastic properites. Changes in membrane elastic properties can occur from changes in membrane composition and forces acting on the membrane. Stability of cell membrane is maintained through a balance of forces acting on the interfacial region and hydrophobic core. Nanoparticles embedded into membrane may cause variation in membrane composition profile, leading to membrane instability. Moreover, interaction between embedded nanoparticles and membrane may affect the forces acting on the membrane and result in membrane deformation. In addition to the membrane deformation, nanoparticles may affect functionality of membrane proteins. An important class of membrane proteins is ion channels which regulate the flow of ions across the membrane in response to various stimuli. The stimuli include electrical signals for voltage-gated ion channels or chemicals in external environment for ligand-gated ion channels. The channel state (open or closed) also depends on lateral pressure profile. Variation of the pressure distribution within the membrane may shift equilibrium balance between open and closed states (Fig. 4-1). 56

57 Nanoparticles embedded into the membrane may affect the pressure profile and, hence, the ion channel functionality. A B Figure 4-1. Lateral pressure within membrane and corresponding different conformational states of a hypothetical membrane protein [6]. Arrows represent the direction and magnitude of lateral pressure. Even if individual nanoparticles introduce a relatively weak perturbation to the membrane, it is possible that aggregates of these nanoparticles will significantly disrupt the membrane. In order to assess this possibility, the tendency of CNTs to aggregate inside the membrane is estimated. This aggregation can be facilitated by a long-range attraction between nanoparticles embedded into the membrane. In the absence of long-range electrostatic interactions between the nanoparticles (which is the case in the current work), the dominant contribution to long-range interactions between the nanoparticles is expected to be mediated by the membrane. Specifically, embedding a CNT into a membrane creates a perturbation of the membrane elastic energy in some neighborhood of the nanoparticle. This perturbation will be modified if another nanoparticle is embedded nearby. The membrane-mediated nanoparticle-nanoparticle interaction will be attractive if decreasing the distance between the nanoparticles reduces the elastic energy of the membrane. In this chapter, effects of nanoparticles on elastic properties of the membrane, namely, bending, tilt, and stretching moduli, and on the lateral pressure profile are investigated in order to consider both direct and indirect effects of nanoparticles on 57

58 the membrane function. SWCNTs are focused on, and three different lengths of the SWCNTs are used to assess the effect of nanoparticle size on the elastic properties. 4.2 Model and Simulation Details CNT are hydrophobic and, therefore, experience a strong driving force toward the hydrocarbon-filled bilayer center. Moreover, the energy barrier for entry of these nanoparticles into the membrane is very small. This property is used to prepare equilibrium bilayers containing CNT. Specifically, nanotubes were placed into a bilayer in size º½ º½ ½½º¼ nm by pushing them with a constant acceleration of 1m/s ¾ toward the bilayer in the direction perpendicular to the bilayer surface. The nanotubes were released before they touched head groups of the bilayer lipids to prevent bilayer disruption due to the pushing force. Nanotube was released when the distance between the nanotube end and the head-group was close to 0.25nm. The released nanotube entered the bilayer within a few nanoseconds. The bilayer containing the carbon nanotube is shown in Fig In order to perform this procedure for nanotubes CNT and CNT, it was necessary to increase the box height to 16 and 21 nm, respectively, to accomodate these nanotubes when they are perpendicular to the bilayer. This was accomplished by adding water molecules to the simulation box. Figure 4-2. Molecular model of a DPPC lipid bilayer containing a carbon nanotube. 58

59 The nm ¾ bilayer systems described will be referred to in the section 2.2 as the small systems. The simulations of these systems were used to assess local perfurbation of the bilayer structure, stress, and energy around the embedded nanotube. However, in order to assess elastic properties of the membrane, it is necessary to consider membrane undulations of sufficiently large wavelength. To this end, simulations of larger ¾ ¾ ¾½ nm systems were performed. These systems will be referred to as the large systems. These systems were obtained by copying laterally the small systems and, if necessary, adding water molecules to ensure that the height all of the large systems is the same, 21 nm. In order to investigate effects of the CNT concentration on the membrane properties, bilayers containing two or five CNT are also considered. These systems were obtrained by removing seven or four nanotubes, respectively, from a large system containing nine CNT. In order to investigate influence of embedded nanotubes on the critical tension γ necessary to rupture the membrane, a series of simulations of a pure membrane is performed in NP Æ γt ensembles. A NP Æ γt ensemble corresponds to constant N, P Æ, γ, and T, where N is a number of molecules, P Æ the the pressure in the direction normal to the membrane surface, γ is the tension in the direction parallel to the membrane surface, and T is the temperature. The Berendsen barostat and the Nosé-Hoover thermostat were employed. The value of the normal pressure È Æ was maintained at 1 bar and the value of the lateral tension γ was maintained at a fixed value between of ¼mN/m and ¼mN/m. These simulations were performed for the small systems. Initial conditions for the simulations with the applied tension was an equilibrated tensionless membrane. The simulations with the tension were performed for 100 ns unless the membrane ruptured within a shorter period of time. 59

60 4.3 System Structure Since the nanotubes are hydrophobic, their preferred location is near the bilayer center. The bilayer density minimum at Þ ¼ provides an additional driving force for CNT towards the bilayer center. This is confirmed by the computed probability distributions of the distance Þ between the nanotube and the bilayer centers of mass and the nanotube orientation Ó θ with respect to the Þ-axis shown in Fig As can be seen, the most likely position of the CNT centers of mass is at or very close to the bilayer center. Moreover, the most likely nanotube orientations are almost parallel to the bilayer surface. Surprisingly, the most likely CNT orientations are not exactly parallel to the bilayer plane, which would minimize the system enthalpy. The deviation from the parallel location can be explained by the entropic contributions to the free energy of the system. As the nanotube length decreases, the enthalpy gained by CNT alignment with the Þ ¼ plane also decreases, while the entropic contribution to the free energy remains roughly the same. Therefore, the most likely orientation of CNT becomes less aligned with the bilayer plane and the fluctuations of CNT orientation become larger as the nanotubes become shorter. The box sizes, Ä Ü, Ä Þ, and average area per lipid, ¼, are summarized in Table 4-1. As can be seen, addition of the nanotube to the system increases the area per lipid and this increase becomes larger for longer nanotubes and higher nanotube concentration. It is interesting to compare the area per lipid in the systems of different sizes containing the same nanotubes at the same concentration. The values of ¼ are almost the same in the small and large systems under the same conditions. However, the larger system typically has a slightly smaller value of ¼, which can be explained by larger magnitude of the bilayer undulations in larger systems. The magnitude of the bilayer undulations increases with the wavelength (see Section 4.4). Therefore, the area (per lipid) of 60

61 P CNT 3 CNT 4 CNT 6 P z (nm) A cos θ B Figure 4-3. Probability distributions of (A) distance Þ between the bilayer and CNT centers of mass and (B) orientation Ó θ of CNT with respect to the Þ-axis. the bilayer projection becomes slightly smaller in larger systems. The only observed exception is the bilayer containing CNT. Table 4-1. Box sizes of equilibrated systems. Æ ÆÌ denotes the number of CNT in the system, Ä Ü Ä Ý is the length of the box side parallel to the bilayer, Ä Þ is the box height, and ¼ is the area per lipid. Statistical errors of these averages are negligibly small, Ç ½¼ µ nm for Ä Ü and Ç ½¼ µ nm for Ä Þ. CNT Æ ÆÌ Ä Ü (nm) Ä Þ (nm) ¼ (nm ¾ ) CNT CNT CNT CNT CNT CNT CNT CNT The observed increase of ¼ with addition of nanoparticles is qualitatively different from observations of Ref. [59]. The latter study showed that addition of fullerenes to a lipid bilayer decreases the area per lipid. 61

62 4.4 Elastic Properties of Membrane The spectral intensities of the membrane undulations and lipid tilt fluctuations obtained for a pure DPPC membrane and DPPC membranes containing CNT of various length are shown in Figure 4-4. Overall, the obtained results are in good agreement with the theoretical prediction Eqs. (2 31),(2 32). However, the following qualitative differences are observed: 1. The magnitude of tilt fluctuations, Ø Õµ ¾, exhibits a weak dependence on the wavenumber Õ, whereas Eq. (2 32) predicts that Ø Õµ ¾ is independent of Õ. 2. The magnitude of the short-wavelength fluctuations of the bending modes scales decays slower as Õ than predicted by the model. Specifically, a least-squares fit of Õµ ¾ for sufficiently large Õ to a power law Õ β results in β < ¾, whereas the model predicts that β ¾ ĥ(q) ĥ(q) Pure DPPC DPPC+CNT q (nm 1 ) ˆt(q) DPPC+CNT DPPC+CNT q (nm 1 ) A q (nm 1 ) B Figure 4-4. Spectral intensity of fluctuation of (A) membrane undulations and (B) lipid tilt in pure DPPC membrane, as well as DPPC membranes containing CNT of various length (see legend) at concentration ¼º¼½ CNT/nm ¾ (i.e. nine CNT per large simulation box). In (A), the main plot shows detail for the long wavelength limit and the inset shows the spectral intensity for all considered wavenumbers. Results obtained using Fourier sums of two different lengths (with five and eleven harmonics) are shown. The dashed lines show results of the fit to the HK model with the cut-off wavenumber Õ Ñ Ü ½º nm ½. There is a systematic dependence of the spectral intensity Õµ ¾ of the bending modes on the length and concentration of nanotubes contained in the membrane. 62

63 Since this dependence is relatively weak, it is necessary to obtain accurate estimates of statistical errors in computed Õµ ¾ to rule out the possibility that the observed differences can be attributed to statistical uncertainties. Since the membrane fluctuations are slow in comparison with the sampling frequency of this MD simulations, the instantaneous normal modes used in the averaging are correlated in time. The autocorrelation functions of the normal modes decay exponentially which indicates that evolution of each of the modes can be described by a first order stochastic differential equation. The Maximum Likelihood (ML) method [46] is used, to obtain the parameters of this equation and, in particular, the magnitude of the membrane fluctuations and the error estimates of this magnitude. The details of the ML method applied to the current system are discussed in Section The obtained error estimates are shown in Fig. 4-4 by error bars. The statistical errors are very small and some of the error bars are smaller than the symbols used to plot the data. Therefore, the observed differences in the magnitude of the membrane fluctuations are significantly larger than the statistical errors. Another possible source of the observed differences in the fluctuation magnitudes is sensitivity of the computed Fourier coefficients to the number of harmonics Æ used in the Fourier sum. Since these coefficients are obtained using least-squares fits to MD data, the values of Õµ and Ø Õµ leading to the best fit may be different for different number of terms included into the sums (2 33). To assess possible effect of Æ on the Fourier modes, Fourier coefficients corresponding to Æ ranging from 5 to 11 were obtained. It is observed that instantaneous values of the Fourier modes Õ Øµ and Ø Õ Øµ are relatively insensitive to Æ. The effect of Æ on the average spectral intensities Õµ ¾ and Ø Õµ ¾ can be gauged from Fig In this figure, the spectral intensities corresponding to Æ and Æ ½½ are plotted. The difference between the bending modes obtained with different Æ is invisible in the plot, which implies that the observed dependence of the spectral 63

64 intensities Õµ ¾ on the length and concentration of CNT embedded in the bilayer cannot be attributed to the details of this numerical procedure. On the other hand, the difference between the tilt modes of the same system but computed with different Æ is larger than the difference between these modes computed for different systems but with the same Æ. Therefore, within the resolution of the current model, the magnitude of the tilt fluctuations appears to be insensitive to the presence of CNT. As it was discussed earlier, there is a systematic deviation of the observed MD data from the HK model predictions for short-wavelength bending modes, Õµ ¾ Õ β, β < ¾ as Õ. This implies that the bending and tilt moduli obtained by fitting the MD data to Eq. (2 31) are sensitive to the upper limit Õ Ñ Ü of the wavenumbers over which the fitting is performed. This is evident from Fig. 4-5 which shows the dependence of κ and κ Ø on Õ Ñ Ü. Clearly, dependence of the elastic moduli on Õ Ñ Ü is much stronger than their dependence on the presence and length of CNT. However, it is observes that for the same value of Õ Ñ Ü, the bending moduli κ Õ Ñ Ü µ increase with CNT length. Moreover, κ Õ Ñ Ü µ appears to have an asymptote as Õ Ñ Ü ¼ and therefore the limit of κ Õ Ñ Ü µ in a sufficiently large system is well defined. The validity of HK model can be also assessed by computing κ Ø using the magnitude of the tilt fluctuations, see Eq As it was discussed earlier, the values of Ø Õµ ¾ obtained from MD simulations show a systematic dependence on Õ, whereas HK model predicts that the magnitude of tilt fluctuations is independent of Õ. Since the dependence of Ø Õµ ¾ on Õ is relatively mild (see Fig. 4-4), it is computed κ Ø using the magnitude of Ø Õµ ¾ averaged over the entire range of wavenumbers Õ Õ Ñ Ü. As Fig. 4-5 shows, the values of κ Ø obtained using this method are somewhat larger than those obtained from the bending modes. This implies that the magnitude of the tilt fluctuations inferred from the bending modes is larger than Ø Õµ ¾ measured directly. This can be explained by additional contributions of the protrusion tension [30] to the 64

65 7 x Pure DPPC DPPC+CNT DPPC+CNT κ (J) 4 3 DPPC+CNT 6 κt (N/m) q (nm 1 ) max A q (nm 1 ) max B Figure 4-5. Dependence of the HK model parameters (A) κ and (B) κ Ø on the upper cut-off wavelength Õ Ñ Ü. The solid lines and filled symbols show the moduli obtained from the bending modes, see Eq. (2 31) and the dashed lines and unfilled symbols show the tilt moduli estimated from the tilt modes, see Eq. (2 32). bending modes at short wavelength. Since the protrusion tension is associated with the motion of the lipid molecules in the direction normal to the bilayer surface, this motion does not contribute to the tilt fluctuations. It is also noted that κ Ø Õ Ñ Ü µ obtained from the tilt modes approaches an asymptote as Õ Ñ Ü ¼, whereas κ Ø Õ Ñ Ü µ obtained from the bending modes does not appear to approach a limit. 4.5 Effect of CNT on Pressure Distribution inside Membranes In addition to changes in the membrane elasticity, nanoparticles embedded in a membrane may affect the membrane function by changing activity of membrane proteins. This can be achieved, e.g., by modification of the lateral pressure inside the membrane, since some membrane proteins, such as ion channels, are sensitive to changes in this pressure. [4, 29, 52]. Therefore, nanoparticle-induced changes in the lateral pressure may shift the equilibrium balance between open and closed states of the channels. 65

66 Pressure tensor is anisotropic inside the membrane. Normal component of the pressure tensor, È È ÞÞ, is position-independent and coincides with the pressure in bulk water to ensure mechanical stability of the membrane. However, the lateral component of the pressure tensor, È È ÜÜ È ÝÝ µ/¾, undergoes significant variations inside the membrane. This can be seen from the profiles of the lateral pressure along the direction normal to the bilayer shown in Fig È is negative at the interface between the hydrophilic head groups and hydrophobic tail groups (Þ ±½º nm) due to interfacial tension between the head- and tail-groups. È is positive in the hydrophobic core and the head-group region of the membrane. The positive pressure is caused by excluded volume repulsion and, in the head-group region, by electrostatic interactions. The balance between the tension and repulsion maintains zero net surface tension of the membrane P (bar) Pure DPPC DPPC + CNT 3 DPPC + CNT 4 DPPC + CNT z (nm) Figure 4-6. Distributions of lateral pressure È in direction normal to the bilayer surface. Pressure profiles inside a pure DPPC bilayer and inside bilayers containing CNT of various lengths are shown. The obtained lateral pressure profile for the pure bilayer is very similar to that obtained by Marrink et al [28] using the same coarse grained model as in this simulations. However, there are some differences between these results and Ref. [28]. Specifically, the magnitude of the minimum pressure at the head-tail interface computed 66

67 in the current work is approximately 120 bar larger than that computed in Ref. [28]. In addition, the pressure maximum in the bilayer center computed in this work is approximately 40 bar smaller. These discrepancies are attributed to the difference in methods of calculation of the stress profile. Marrink et al. have used the method described in Ref. [24]. In this method, the assignment of the weight function for a slab containing one of the end-points of the IK contour is performed less accurately than in the current work. Specifically, in [24] location of the bead contained in the slab is approximated as located exactly halfway between the slab boundaries. In the current work, precise bead locations in the calculation of the weight function are used. Although the difference of methods is relatively small, the gradient of the stress profile is very large around the stress extrema. Therefore, it is possible that this difference accounts for the quantitative discrepancy between this result and that of Ref. [28]. Addition of CNT to the membrane does not change the qualitative shape of the pressure profile but does lead to quantitative changes in the lateral pressure profile, as can be seen from Fig Pressure becomes larger in the bilayer center due to excluded-volume interactions between the lipid tail-groups and CNT. Increasing the CNT length leads to an increased deviation of the pressure profile from that in a CNT-free membrane. However, the distribution of the lateral pressure in the membrane plane shown in Fig. 4-7 demonstrates that the changes in the lateral pressure are localized to the immediate neighborhood of the embedded CNT. Therefore, these nanotubes are unlikely to have significant impact on the membrane protein function. 4.6 Membrane Energy around Nanoparticles When the nanoparticle is localized at its favorable position inside the membrane, the membrane remains deformed. Mechanical stress created by these membrane deformations is directly responsible for membrane-mediated interactions between the membrane inclusions (such as nanoparticles and proteins). For example, two membrane 67

68 y (nm) y (nm) x (nm) A x (nm) B y (nm) x (nm) C Figure 4-7. Distributions of lateral pressure È (in bar) in the bilayer plane for bilayers containing (A) CNT, (B) CNT, and (C) CNT nanotubes. The origin of the system of coordinates coincides with the location of the CNT center of mass and the Ü-axis is aligned along the nanotube. inclusions would be attracted to each other if decrease of the distance between them leads to a decrease in the total stress within the membrane. In the current section, the membrane deformation and the corresponding energy penalty caused by nanoparticles located at their preferred positions within the membrane are assessed. The membrane energy is estimated using the Hamm-Kozlov model (2 30). To apply this model, the mean dividing surface Ü Ý µ and lipid tilt Ø Ü Ý µ 68

69 for each of the bilayer leaflets are obtained. Results of these calculation for bilayers containing various nanotubes are shown in Fig. 4-8, 4-9, and The system of coordinates used in these calculations was chosen so that (i) the origin coincides with the projection of the CNT center of mass on the Ü Ý plane; (ii) the CNT center of mass is located in the upper half-space, (iii) the Ü-axis coincides with the CNT axis of symmetry, and (iv) CNT points upward in the positive direction of the Ü-axis. As can be seen from Fig. 4-8, 4-9, 4-10, nanotubes creates a membrane protrusion around the CNT end pointing towards the dividing surface. In addition, a relatively small deviation of lipid tilt from its bulk (zero) value is observed. The largest magnitudes of the tilt vectors are For CNT : 0.26 (upper leaflet) and 0.17 (lower leaflet), For CNT : 0.19 (upper leaflet) and 0.15 (lower leaflet), For CNT : 0.20 (upper leaflet) and 0.16 (lower leaflet), Since the perturbations to both the membrane shape and lipid tilt are localized to the immediate neighborhood of the nanotubes, the perturbation of the elastic energy of the membrane will also be localized. This result is consistent with the localized perturbation to the lateral pressure discussed in the previous section. Therefore, interactions between CNT and other membrane inclusions are expected to be rather weak and short-ranged. 4.7 Conclusions The following effects of nanoparticles on the membranes were observed: Carbon nanotubes embedded in lipid membranes lead to the membrane softening, which becomes more significant with increase of the CNT length and concentration. Inclusion of carbon nanotubes into a membrane leads to perturbation of the lateral pressure profile within the membrane. However, this perturbation is localized and is unlikely to affect function of membrane proteins. 69

70 A B Figure 4-8. Effect of CNT embedded in a DPPC membrane on (A) upper and (B) lower membrane leaflet. The height of the dividing surface is shown by the color plot (in nm); the lipid tilt are shown by the arrows. The arrows are scaled to have consistent scaling between each other; otherwise the length of the arrows in the plot is arbitrary. Carbon nanotubes located at their equilibrium positions inside a lipid membrane introduce relatively small and localized perturbations to the membrane energy. This implies that interactions between these nanoparticles and other membrane inclusions (such as membrane proteins) are relatively weak when the nanoparticles are located at their equilibrium positions. The interactions of the nanoparticle are out of reach to other nanoparticles, which is consistent with other simulations showing that fullerenes do not form stable aggregates inside the bilayer [59]. 70

71 A B Figure 4-9. Effect of CNT embedded in a DPPC membrane on (A) upper and (B) lower membrane leaflet. See Fig. 4-8 for details. A B Figure Effect of CNT embedded in a DPPC membrane on (A) upper and (B) lower membrane leaflet. See Fig. 4-8 for details. 71

72 CHAPTER 5 LIPID PEROXIDATION 5.1 Introduction In biological cells, oxidative stress occurs due to an imbalance between reactive oxygen species (ROS) produced by normal aerobic metabolism and the ability of cells to recover from the resulting damage. The produced ROS may damage such biomolecules as nucleic acids, proteins, and lipids [37]. In particular, the lipid peroxidation is the process of the oxidative degradation of lipids and occurs when free ROS radicals take electrons from the lipids in cell membranes. One of the mechanisms of the lipid peroxidation in which lipid molecules are attacked by free radicals was proposed by Wang et al. [57]. ÄÀ ÇÀ Ä À ¾ Ç (5 1) Ä Ç ¾ ÄÇÇ (5 2) Ä ÇÇ ÄÀ Ä ÄÇÇÀº (5 3) Here, ÄÀ denotes lipid with hydrocarbon chain (-CH ¾ ) and hydrogen (H) and ÄÇÇÀ denotes a lipid peroxide. If the lipid peroxidation reaction is not terminated fast enough, the cell membrane will be damaged [18]. Several groups have argued that the nanoparticle toxicity can be attributed to the oxidative stress and the lipid peroxidation. E.g. the pioneering work of Oberd ørster reported the oxidative stress and lipid peroxidation in the brain of a large mouth bass following exposure to fullerenes [36]. In addition, Sayes et al. [43] observed the disruption of cell membrane exposed to C ¼. It was shown that C ¼ induces production of the superoxide anion (O ¾ ) in cell-free aqueous solutions. The superoxide anions can generate a hydroxyl radical ( ÇÀ) as a product of the reaction between the superoxide anions and hydrogen peroxide (H ¾ O ¾ ) as follows: Ç ¾ À ¾Ç ¾ ÇÀ ÇÀ Ç ¾ º (5 4) 72

73 The produced hydroxyl radical accelerates the production of peroxidized lipids as shown in the reactions (5 1), (5 2), and (5 3). Although the fullerenes do not directly participate in the shown lipid peroxidation reactions, it was demonstrated experimentally that they increase concentration of ROS in water and induce disruption of cell membranes. CNTs may also be able to induce the lipid peroxidation. Electron spin resonance of SWCNT-stimulated human epidermal keratinocytes cells provides evidence of accumulation of peroxidation products and a decrease of intracellular levels of glutathione, a major natural antioxidant [47]. Transmission electron microscopy images of the cell membranes exposed to SWCNT show morphological changes. In contrast with the large number of experimental studies of peroxidation of lipid membranes, there is currently a very limited number of computational studies of this phenomenon. The first MD study of effect of lipid oxidation on the structural properities of model membranes was performed by Wong-Ekkabut et al. [60]. The authors investigated effect of four different products of 1-palmitoyl-2-linoleoyl-sn-glyceero-3-phosphatidylcholine (PLPC) peroxidation on PLPC bilayers at five concentrations, ranging from 2.8% to 50% using an atomistic model. It is observed that oxidized functional groups in the lipid tails cause the tails to bend toward the aqueous phase and form hydrogen bonds with water and the lipid head groups. This internal structural change of the bilayer results in the increase of the average area per lipid and the decrease of the bilayer thickness in agreement with experimental results [40, 49]. Moreover, the tendency of the oxidized lipid tails to bend toward the water phase increases the membrane permeability by water. The study shows not only the significant changes in the structure of membrane but also the possibility of membrane rupture indicated by formation of water pores. Possible impacts of oxidized lipids on membrane-associated biological process have been examined by MD simulations of Khandelia and Mouritsen [19]. They evaluated structural changes in the palmitoyloleoyl phosphatidylcholine (POPC) lipid bilayer due to 73

74 the presence of two types of oxidized lipids, zwittterionic PoxnoPC and anionic PazePC. It was observed that the anionic functional groups prefer to reside in the aqueous phase and their negative charge would attract positively charged peptides, ions, drugs, and hormones. While computational studies of the lipid peroxidation have focused on changes of structure of oxidized membranes, to the best of the knowledge, effects of the presence of nanoparticles on the peroxidized membranes have never been investigated using computational methods. In the current study, membrane instability due to a combination of lipid peroxidation and a nanoparticle present in the membrane is investigated. 5.2 Model and Simulation Details The peroxidized lipid LOOH in reaction (5 3) is modeled by replacing the hydrophobic bead at the end of one of the lipid tails by a hydrophilic bead with polarity lower than that of water. The hydrophilic peroxidized bead has a strong polar interaction with the negatively charged bead (ǫ = 5.0 kj/mol) and less polar interaction with the positively charged bead (ǫ = 4.0 kj/mol) and glycerols and water (4.5 kj/mol). It has a hydrophobic repulsion with carbon beads in tails (ǫ = 2.7 kj/mol). The bond and angle potentials for the peroxidized lipid are the same as those for DPPC lipids. Reaction (5 3) suggests that both of the lipid tails are equally likely to randomly replace their end bead by a hydroperoxide group (-OOH). Therefore, the system is prepared as follows. Random lipid molecules in the pure DPPC bilayer ( º½ º½ ½½º¼ nm ) are replaced by the peroxidized lipids at three concentrations (25%, 50%, and 75%), keeping the number of the peroxidized lipids equal in both lipid monolayers. Then each system is copied laterally to create a new bilayer system of nm. The obtained systems are equilibrated for 50ns then simulated for 800ns. In what follows, these peroxidized bilayers containing 25%, 50%, and 75% peroxidized lipids are denoted as DPPC-25%, DPPC-50%, and DPPC-75%, respectively. In addition, simulation of peroxidized bilayers containig a fullerenol is 74

75 performed in order to investigate if the nanoparticle presence changes the stability of the peroxidized membrane. A pure DPPC bilayer containing a fullerenol in a simulation box of nm (592 DPPC molecules and water beads) is prepared. Then random lipid molecules are replaced by the peroxidized lipids in the pure bilayer containing the particle. The systems was equilibrated for 100ns followed by a production run for 200ns. 5.3 Effect of Lipid Peroxidation on Membrane Properties Structural changes are observed in bilayers containing peroxidized lipids. Snapshots of the bilayers containing peroxidized and non-peroxidized lipids at equilibrium are shown in Fig The peroxidized bilayers undergo fluctuations of higher magnitude in comparison with the pure bilayer. The difference between the fluctuations in the peroxidized bilayers and pure bilayers increases as the concentration of peroxidized lipids increases. In addition, the bilayer with the largest concentration of peroxidized lipids (DPPC-75%) undergoes local perturbation of its surface. The dependence of the spectral intensity of the bilayer bending modes on the wavevector q is shown in Fig This plot confirms that the fluctuation magnitude increases as the concentration of peroxidized lipids increases. This refers lipid peroxidation leads to more flexible membranes. Moreover, the dependence of the fluctuation magnitude on the wave number in the bilayers containing high concentration of peroxidized lipids is qualitatively different from that of the pure bilayer. The q-dependence of the fluctuations of the peroxidized bilayers exhibits a sharp transition between the long and short wavelength modes whereas that of the pure bilayer shows a smooth transition between the modes of all wavelengths. The transition is also smooth in the bilayer with the lowest considered concentration of peroxidized lipids (DPPC-25%). In order to examine the structural changes in detail, a number of properties of a DPPC bilayer is investigated. The density profiles of the pure and peroxidized bilayers are shown in Fig. 3-1A. As shown in Fig. 5-3A and 5-3B, densities of the head- and tail-groups in the peroxidized 75

76 A B C D Figure 5-1. Simulation snapshots of DPPC bilayers at 800ns (A) 0% (pure DPPC lipid bilayer), (B) DPPC-25%, and (C) DPPC-50%, and (D) DPPC-75%. bilayers have broader distributions than those in the pure bilayer. A fraction of the head groups in peroxidized bilayers is located at the bilayer center, and some tail groups in these bilayers are located in the water phase. The peroxidized beads are attracted to head-groups of lipids, and hence some of the head-groups permeate into the bilayer core due to this attraction. Water also penetrates the peroxidized bilayer as shown in Fig. 5-3C. Non-negligible amounts of water are present in the tail-rich region of the bilayer and in particular, water is present at the center of the DPPC-75% bilayer. It is possible that the peroxidized lipids induce a formation of water pores. The disruption of the bilayer is more pronounced when the peroxidized lipid concentration is higher. It is observed that some of the water-soluble terminal beads of peroxidized lipids align at the interface. Attraction between the peroxidized terminal beads and water causes a lipid tail-chain containing the peroxidized bead to reorient itself so that the hydroperoxide group points out into the aqueous phase. This can be seen from Fig. 5-4 which shows the density distribution of the peroxidized terminal beads in the bilayer. These beads are distributed from the center of the bilayer up to the interface (-3nm and 76

77 % 25% 50% 75% ĥ(q) q (nm 1 ) Figure 5-2. Spectral intensity of bilayer surface fluctuations of the pure DPPC bilayer and of DPPC-25%, DPPC-50%, and DPPC-75%. 3nm) between head-groups of lipids and water. This indicates that a peroxidized tail folds toward the aqueous phase, resulting in internal structural changes in the bilayer. It is shown in Fig. 5-5 that the higher concentration of peroxidized lipids leads to a broader distribution and higher density of the peroxidized lipids at the bilayer center. The distribution becomes broader because the bilayer in the higher peroxidized lipid concentration is more fluctuating and disrupted. To gain further insight, the fraction of peroxidized DPPC lipids with tails folded towards the water phase is obtained. A tail to be folded is considered if the position of the replaced peroxidized bead (P) is further away from the bilayer center than the tail bead (C1 or C5) connected to GL1 or GL2 (see Fig. 5-6). Table 5-1 shows dependence of the average fraction of the folded lipids on the concentration of peroxidized lipids in the bilayers. Since the polarity of the peroxide groups is lower than water, not all of the tails are folded toward water (see Fig. 5-5). The fraction of the folded lipids 77

78 Bead density (beads/nm 3 ) % 25% 50% 75% Bead density (beads/nm 3 ) % 25% 50% 75% z (nm) A z (nm) B 10 Bead density (beads/nm 3 ) % 25% 50% 75% z (nm) C Figure 5-3. Density profiles of head and tail groups of lipid and water in the pure DPPC bilayer, DPPC-25%, DPPC-50%, and DPPC-75%. decreases as the concentration of peroxidized lipids increases. When the peroxidized lipid concentration is low, there are fewer hydroperoxide beads in an immediate neighborhood of each of the peroxidized lipid, i.e. peroxidized tails are likely to be surrounded by hydrophobic tail groups unless they fold towards the head group. On the other hand, when the concentration of peroxidized lipids increases, a peroxidized lipid is likely to be in contact with another peroxidized lipid. Therefore, the peroxidized beads will attract each other in the core of the bilayer and the driving force towards tail 78

79 Bead density (beads/nm 3 ) PO 4 Glycerol CH 3 COOH z (nm) Figure 5-4. Density profiles of the phosphate beads (PO, dotted) of the head group, the glycerol (dash-dotted), the terminal beads (-CH, dashed), and the peroxidized terminal beads (-COOH, solid) in DPPC-25%. Bead density (beads/nm 3 ) % 50% 75% z (nm) Figure 5-5. Density profiles of the peroxidized terminal beads in DPPC-25% (dotted), DPPC-50% (dash-dotted), and DPPC-75% (solid). folding will decrease. Thus, the fraction of folded tails in the case of high concentration is smaller than in the case of low concentration of peroxidized lipids. The changes in the tail conformations lead to changes in the average area per lipid in bilayer containing peroxidized lipids. The areas per lipid in the peroxidized bilayers in Table 5-1 are larger than that in the pure bilayer (0.6356nm ¾ /lipid), and a further 79

80 deviation from the pure bilayer is observed as the concentration of the peroxidized lipids increases. The presence of peroxidized terminal beads at the interface in the peroxidized lipid bilayer causes a repulsion between the peroxidized beads and head-groups due to the excluded volume interactions, which leads to the increase in the surface area per lipid. More terminal beads at higher peroxidized lipid concentration induces more excluded volume. Figure 5-6. Peroxidized DPPC lipid with folded tails. NC and PO refer to a positively and negatively charged beads, respectively. GL1 and GL2 refer to glycerols. C and P refer to hydrophobic carbon bead and a peroxidized bead. Table 5-1. The average fraction of folded lipids and the average area per lipid in bilayers with various concentrations of peroxidized lipids. Concentration Fraction of folded tails Area per lipid (nm ¾ /lipid) 25% % % Effect of Nanoparticles on Peroxidized Lipid Bilayers As discussed in Chapter 3.2, morphological changes may be caused in bilayer containing nanoparticles due to physical interaction between the nanoparticles and the 80